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BALTIMORE LECTURES 

ON 

MOLECULAR DYNAMICS 

AND 

THE WAVE THEORY OF LIGHT 



Eonfcon: C. J. CLAY AND SONS, 

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 

50, WELLINGTON STREET. 




Baltimore: PUBLICATION AGENCY OF THE JOHNS HOPKINS UNIVERSITY. 

Bombag anU Calcutta: MACMILLAN AND CO.. LTD 

l(ip}ig: F. A. BROCKHAUS. 



[All nights reserved.] 



BALTIMORE LECTUKES 

ON 

MOLECULAE DYNAMICS 

AND 

THE WAVE THEOEY OF LIGHT 

FOUNDED ON ME A. S. HATHAWAY'S STENOGRAPHIC REPORT OP 
TWENTY LECTURES DELIVERED IN JOHNS HOPKINS 

UNIVERSITY, BALTIMORE, IN OCTOBER 1884: 
FOLLOWED BY TWELVE APPENDICES ON ALLIED SUBJECTS 



Will i fin "DiflrynStrn 
LORD KELVIN, O.M., G.C.V.O., P.O., F.R.S., &c. 

PRESIDENT OF THE ROYAL SOCIETY OF EDINBURGH, 

FELLOW OF 9T PETER'S COLLEGE, CAMBRIDGE, 
AND EMERITUS PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF GLASGOW. 



LONDON: 

C. J. CLAY AND SONS, 

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE, 

AVE MARIA LANE. 

1904 



fK.I, 



y 









Ombi ifegr : 

PBINTKD BT J. AND C. P. CLAY, 
AT THE CNIVER8ITT PRESS. 



PREFACE. 



ERRATA. ISSUED 1906. 



ERRATUM. 
Page 116, line 13 from foot, for " IT" read " tri. 



700, line 14, for "Marlowe" read "Marloye." 



We all felt that difficulties were to be faced and not to be 
evaded ; were to be taken to heart with the hope of solving 
them if possible ; but at all events with the certain assurance 
that there is an explanation of every difficulty though we may 
never succeed in finding it. 



PREFACE. 



ERRATA, ISSUED 1906. 

Page 53, line 6 from foot, for "Marlowe" read "Marloye." 

Pages 72, 74, 76, 78, 82, 84, 86, 88, 90, 92, head lines, delete "Part I. 

Page 100, line 16 from foot, for "Lommell" read "Lommel." 

109, last line, for ".4 cos..." read "acos...." 

394, margin, for "raloM" read "Molar." 

,, 413, line 6 from foot, for "D" read "." 

506, line 11, transpose U^ and U 2 . 

528, equations (3) and (4), for "cosvt" read "cosqvt." 

546, last line of 9, for "2-27" read "2.27." 

549, footnote, line 3, for "^" read u ~." 

559, line 10 of 31, delete "vitreous and resinous." 
700, line 14, for "Marlowe" read "Marloye." 



We all felt that difficulties were to be faced and not to be 
evaded ; were to be taken to heart with the hope of solving 
them if possible ; but at all events with the certain assurance 
that there is an explanation of every difficulty though we may 
never succeed in finding it. 



PREFACE. 



HAVING been invited by President Oilman to deliver a 
course of lectures in the Johns Hopkins University after 
the meeting of the British Association in Montreal in 1884, on 
a subject in Physical Science to be chosen by myself, I gladly 
accepted the invitation. I chose as subject the Wave Theory 
of Light with the intention of accentuating its failures ; rather 
than of setting forth to junior students the admirable success with 
which this beautiful theory had explained all that was known 
of light before the time of Fresnel and Thomas Young, and 
had produced floods of new knowledge splendidly enriching the 
whole domain of physical science. My audience was to consist of 
Professorial fellow-students in physical science ; and from the 
beginning I felt that our meetings were to be conferences of 
coefficients, in endeavours to advance science, rather than teach- 
ings of my comrades by myself. I spoke with absolute freedom, 
and had never the slightest fear of undermining their perfect 
faith in ether and its light-giving waves : by anything I could 
tell them of the imperfection of our mathematics ; of the insuffi- 
ciency or faultiness of our views regarding the dynamical qualities 
of ether ; and of the overwhelmingly great difficulty of finding 
a field of action for ether among the atoms of ponderable matter. 
We all felt that difficulties were to be faced and not to be 
evaded ; were to be taken to heart with the hope of solving 
them if possible ; but at all events with the certain assurance 
that there is an explanation of every difficulty though we may 
never succeed in finding it. 



PREFACE. 



It is in some measure satisfactory to me, and I hope it will 
be satisfactory to all my Baltimore coefficients still alive in our 
world of science, when this volume reaches their hands; to find 
in it dynamical explanations of every one of the difficulties with 
which we were concerned from the first to the last of our twenty 
lectures of 1884. All of us will, I am sure, feel sympathetically 
interested in knowing that two of ourselves, Michelson and Morley, 
have by their great experimental work on the motion of ether 
relatively to the earth, raised the one and only serious objection* 
against our dynamical explanations; because they involve the 
assumption that ether, in the space traversed by the earth and 
other bodies of the solar system, is at rest absolutely except in so 
far as it is moved by waves of light or radiant heat or variations 
of magnetic force. It is to be hoped that farther experiments 
will be made; to answer decisively the great question: is, or 
is not, ether at rest absolutely throughout the universe, except 
in so far as it is moved by waves generated by motions of ponder- 
able matter? I cannot but feel that the true answer to this 
question is in the affirmative, in all probability : and provisionally, 
I assume that it is so, but always bear in mind that experimental 
proof or disproof is waited for. As far as we can be contented 
with this position, we may feel satisfied that all the difficulties 
of 1884, set forth in Lectures I, X, and XV, are thoroughly 
explained in Lectures XVIII, XIX, and XX, as written afresh 
in 1902 and 1903. 

It seems to me that the next real advances to be looked for in 
the dynamics of ether are : 

(I) An explanation of its condition in the neighbourhood of 
a steel magnet or of an electromagnet ; in virtue of which mutual 
static force acts between two magnets whether in void ether or in 
space occupied also by gaseous, liquid, or solid, ponderable matter. 

(II) An investigation of the mutual force between two moving 
electrions, modified from purely Boscovichian repulsion; as it must 
be by the composition, with that force, of a force due to the inertia 

* See Appendix A 18 and Appendix B 10. 



PREFACE. vii 

of the ether set in motion by the motion of each of the electrions. 
It seems to me that, of these, (II) may be at present fairly within 
our reach ; but that (I) needs a property of ether not included 
in the mere elastic-solid-theory worked out in the present volume. 
My object in undertaking the Baltimore Lectures was to find how 
much of the phenomena of light can be explained without going 
beyond the elastic-solid-theory. We have now our answer : every 
thing non-magnetic; nothing magnetic. The so-called "electro- 
magnetic theory of light " has not helped us hitherto : but the 
grand object is fully before us of finding a comprehensive dynamics 
of ether, electricity, and ponderable matter, which shall include 
electrostatic force, magnetostatic force, electromagnetism, electro- 
chemistry, and the wave theory of light. 

I take this opportunity of expressing the gratitude with which 
I remember the hearty and genial cooperation of my coefficients 
in our meetings of 19 years ago in Baltimore, and particularly 
the active help given me by the late Prof. Rowland, from day to 
day all through our work. 

I desire also to specially thank one of our number, Mr A. S. 
Hathaway, for the care and fidelity with which he stenographically 
recorded my lectures, and gave his report to the Johns Hopkins 
University in the papyrograph volume published in December 1884. 
The first eleven lectures, as they appear in the present volume, 
have been printed from the papyrograph, with but little of even 
verbal correction ; and with 'a few short additions duly dated. 

Thirteen and a half years after the delivery of the lectures, 
some large additions were inserted in Lecture XII. In Lectures 
XIII, XIV, XV, freshly written additions supersede larger and 
larger portions of the papyrograph report, which still formed 
the foundation of each Lecture. Lectures XVI XX have been 
written afresh during 1901, 1902, 1903. 

In my work of the last five years for the present volume 
I have received valuable assistance successively from Mr W. Craig 
Henderson, Mr W. Anderson, and Mr G. A. Witherington ; not 

a5 



Viii PREFACE. 

only in secretarial affairs, but frequently also in severe mathe- 
matical calculation and drawing; and I feel very grateful to them 
for all they have done for me. 

The printing of the present volume began in August, 1885 ; 
and it has gone on at irregular intervals during the 19 years 
since that time; in a manner which I am afraid must have 
been exceedingly inconvenient to the printers. 

I desire to thank Messrs J. and C. F. Clay and the Cambridge 
University Press for their never-failing obligingness and efficiency 
in working for me in such trying circumstances, and for the 
admirable care with which they have done everything that could 
be done to secure accuracy and typographical perfection. 



KELVIN. 



NETHERHALL, 

January, 1904. 



CONTENTS. 

LECTURE I. 

INTRODUCTORY. 

The wave theory of light ; molecular treatment by Fresnel and 
Cauchy ; molar by Green ........ 

Ordinary dispersion. Anomalous dispersion ; hypotheses of Cauchy 
and Helmholtz. Time-element in constitution .of matter. 
" Electromagnetic Theory of Light." Nature of ether ; tenta- 
tive comparison with shoemakers' wax ..... 

Direction of the vibrations in polarized light. Dynamical theory 
of refraction and reflection not complete ; differences of rigidity 
or of density of the ether on the two sides of a reflecting 
interface ; experiments of Prof. Rood on reflectivity 

Double refraction ; its difficulties. Explanations suggested by 
Raukine and by Rayleigh tested by Stokes ; and Huyghens' 
Law verified. Further illustration by mechanical model. 
Conclusion . ... 



LECTURE II. 
PART I. 

Molar. Dynamics of elastic solid. James Thomson's radian. General 
equations of energy and force. Equations for isotropic material. 
Relation between Green's twenty-one coefficients and the bulk 

and rigidity moduluses 

General equations of motion ; modification for heterogeneousness ; 
case of zero rigidity ; compressible fluid. Lord Rayleigh's work 
on the blue sky 

PART II. 

Molecular. Dynamics of a row of connected particles. Illustrative mechanical 
model. The propagation of light waves through the atmosphere. 
Equations of motion 



CONTENTS. 
LECTURE III. 

Molar. Dynamics of elastic solid. Illustrative model for twenty-one 
coefficients. Equations of motion ; methods of solving. Velc 
cities of the condensational and distortional waves. Leslie's 
experiment on sound of a bell in hydrogen explained by Stokes 34-38 

PART II. 
Molecular. Variations of complex serial molecule. Continued fraction to aid 

solution * ' 

LECTURE IV. 

Molar. Equations of motion of elastic solid ; method of obtaining all possible 
solutions. Energy of condensational waves of ether negligible. 
"Electromagnetic theory of light" wants dynamical founda- 
tion ; propagation of electric and magnetic disturbances to be 
brought within the wave theory of light . 

Solutions for condensational waves travelling outwards from a point ; 
spherical waves ; sphere vibrating in ether ; two spheres 
vibrating oppositely in line of centres ; aerial vibrations round 
tuning fork ; cone of silence . . . . . . 46-51 

LECTURE V. 

Molar. Vibrations of air around a tuning fork continued ... 52 

Molecular. Vibrations of serial molecule. Period and ratios of displacement ; 

critical periods ; infinite period 53-58 

Fluorescence. Finite sequences of waves in dispersive medium ; 

beginnings and endings of light 58-60 

LECTURE VI. 

Molar. Ratio of rigidity to compressibility. Velocity -potential. Conden- 
sational wave ; vibrations at a distance from origin ; circular 
bell, tuning fork, and elliptic bell .. 61-65 

Vibrations of approximately circular plates ; planes of silence. 

Velocity of groups of waves through transparent substances . 65-68 

Molecular. Vibrations of serial molecule. Lagrange ; " algorithm of finite 

differences " ; number of terms in determinant . . . 69-70 

LECTURE VII. 

Molecular. Vibrations of serial molecule. Solution in terms of roots ; deter- 

mination of roots , . . . . . . 71-76 

Metallic reflection; Rayleigh on Cauchy ; Sellmeier, Helmholtz, 
Lommel. Lommel on double refraction, interesting but not 
satisfactory ; Stokes' experimental disproof of Rankine's theory 
[if ether wholly incompressible. See Lecture xix, 184, below] 76-79 



CONTENTS. 



LECTURE VIII. 

Molar. Solutions for distortional waves. Rotational oscillation in origin ; 
illustration by wooden ball in jelly. Double source with 
contrary rotations. Solution for waves generated by to-and-fro 
vibrator ; Stokes, Rayleigh, an(* the blue of the sky 

Molecular. Sudden and gradual commencements of vibration ; fluorescence 
and phosphorescence ; initial and permanent refraction ; 
anomalous dispersion fading after a time . . . . 

LECTURE IX. 

Molar. Contrary vibrators in one line. Sudden beginning and gradual 
ending of vibrations in a source of light. Observations on 
interference ; Sellmeier, Fizeau. Conference with Prof. Row- 
land ; conclusion that subsidence of vibrations of molecular 
source is small in several hundred thousand vibrations ; 

dynamics invoked 

Amount of energy in the vibration of an atom. Sellmeier, 
Helmholtz, and Lommel, on the loss of energy in progress 
of waves. Dynamics of absorption ; first taught by Stokes. 
Sellmeier's dynamics of anomalous dispersion 

Molecular. Problem of seven vibrating particles. Dynamical explanation 
of ordinary refraction ; mechanical model ; expression for 
refractive index 

LECTURE X. 

Molar. Energy of waves. Name for y 2 ; Laplacian preferred. Geometrical 
illustrations of Fourier's theorem for representation of 

arbitrary functions . . 

Cauchy and Poisson on deep-sea waves j great contest of 1815, 
Cauchy or Poisson to rule the waves 

Molecular. Difficulties regarding polarization by reflection, double refraction ; 
form of wave-surface. Anomalous dispersion, fluorescence, 
phosphorescence, and radiant heat, discoverable by dynamics 
alone. Radiant heat from Leslie cube. Refractivity through 
wide range. Refractivity of rock-salt ; Langley . . . 

LECTURE XI. 

Molar. " Anisotropy" rejected ; aeolotropy suggested by Prof. Lushington. 
Aeolotropy in an elastic solid. Father Boscovich judged 
obsolete in 1884 [reinstated as guide in 1900. See Lectures xv 
to xx and Appendices passim]. Green's twenty-one modu- 
luses of elasticity. Navier- Poisson ratio ; disproof, and 
Green's full theory illustrated by mechanical models. Most 
general plane wave expressed in terms of the twenty-one 
independent moduluses . . . . . . . . 



Xll 



CONTENTS. 



LECTURE XII. 

PART I. 

Molar. Three sets of plane waves with fronts parallel to one plane ; ^wave- 
surface with three sheets ; Blanchet ; Poisson, Coriolis, Sturm, 
Liouville, and Duhamel, on Blanchet. Purely distortional 
wave; Green. Incompressibility ; energy of condensational 
waves in ether; density of the luminiferous ether 
PART II. 

Molecular. Mutual force between atom and ether. Vibrating molecule; 
polarized source of light. Dynamics of dispersion ; Sellmeier's 
theory ; refractivity formula, and Rubens' experimental results 
for refractivity of rock-salt and sylvin. "Mikrom" and 

"michroh" 

Critical periods, /*=!, /* 2 =0, /i 2 passing from -to to + co . Con- 
tinuity in undulatory theory through wide range of frequencies ; 
mechanical, electrical, and electromagnetic vibrations all in- 
cluded in sound and light 

LECTURE XIII. 

Molecular. Prof. Morley's numerical solution for vibrator of seven periods. 
Dynamical wave machine . 

Molar. JSolotropy resumed. Direction of displacement and return force. 
Incompressible seolotropic elastic solid. French classical 
nomenclature. Square acolotropy. Rankine's nomenclature. 
Cubic seolotropy . , . . . . . . . . 

^Eolgtropic elasticity without skewness ; equalisation of velocities 
for one plane of distortion. Green's dynamics for Fresnel's 
wave-surface ' . 

Molecular. Application of Sellmeier's dynamical theory, to the dark lines 
D^ ; to a single dark line. Photographs of anomalous 
dispersion by prisms of sodium-vapour by Henri Becquerel f . 

LECTURE XIV. 

Molecular. Rowland's model vibrator. Motion of ether with embedded mole- 
cules . . , . ... 
Molar. Mathematical investigation of spherical waves in elastic solid 

Stokes' analysis into two constituents, equivoluminal and irrota- 

tional 

Simplification of waves at great distance from origin 
Examples by choice of arbitrary functions . . ' 

Details of motion, in equator, in 45 cone, and in axis of disturbance 
Originating disturbance at a spherical surface S; full solution for 
S rigid 

Diagrams to illustrate motion of incompressible elastic solid 
through infinite space around a vibrating rigid globe . 



135-142 



142-154 



I.-,:. H;L> 



163-165 



165-170 



170-175 



176-184 



185-190 
190-219 

193-194 
197 
199 

200-201 

202-219 
206-207 



CONTENTS. xiii 

PAGES 

Molar. Rates of transmitting energy outwards by the two waves . . 211-214 
Examples 214-219 

LECTURE XV. 

Molecular. Model vibrator; excitation of synchronous vibrators in molecule 

by light 220-221 

Molar. Thlipsinomic treatment of compressibility. Double refraction; 
difficulty ; wave-velocity depends on line of vibration ; Stokes ; 
Cauchy and Green's extraneous force ; stress theory . . 221-240 
Hypothetical elastic solid. Further development of stress theory ; 
influence of ponderable atoms on ether ; Kerr's observations 
unfavourable to the stress theory of double refraction; 
Glazebrook founds successfully on seolotropic inertia . . 241-259 

LECTURE XVI. 

Molar. Mechanical value of sunlight and possible density of ether. 
Rigidity of ether; etherial non-resistance to motion of pon- 
derable matter. Is ether gravitational ? 260-267 
Gravitational matter distributed throughout a very large volume ; 
quantity of matter in the stellar system ; velocities of stars ; 
great velocity of 1830 Groombridge; Newcomb's suggestion. 
Total apparent area of stars exceedingly small. Number of 
visible stars? Masses of stars 267-278 

LECTURE XVII. 

Molecular. Molecular dimensions ; estimates of Thomas Young, W. Thomson, 
Johnstone Stoney, Loschmidt. Tensile strength in black spot 
of soap bubble ; Newton ; its thickness measured by Reinold 
and Rucker. Thinnest film of oil on water; Rontgen and 
Rayleigh 279-286 

Kinetic theory of gases ; molecular diffusivity ; theory of Clausius 
and Maxwell ; O. E. Meyer. Inter-diffusivity of pairs of gases ; 
viscosity of gases; experimental determinations of inter- 
diffusivities by Loschmidt and of viscosities by Obermayer; 
Maxwell 286-298 

Number of molecules in a cubic centimetre of gas estimated. 
Stokes on the polarized light from a cloudless sky ; Rayleigh's 
theory of the blueness of the sky. Transparency of atmo- 
sphere; Bouguer and Aitken. Haze due to larger particles. 
Dynamics of the blue sky 298-313 

Sky -light over Etna and Monte Rosa; observations of Majorana 
and Sella. Definition of Lambert's "albedo." Table of 
albedos; Becker, Muller. Prismatic analysis of sky-light; 
Rayleigh; researches of Giuseppe Zettwuch. Atmospheric 
absorption; Muller and Becker. Molecular dimensions of 
particular gaseous liquid and solid substances . . . 313-322 



xiv 



CONTENTS. 



Molecular. Vibrations of polarized light are perpendicular to plane of polari- 
zation Stokes' experimental proofs ; experiments of Holtzman 
and Lorenz of Denmark; confirmation of Stokes by Rayleigh 
and Lorenz 



PAGES 



LECTURE XVIII. 

REFLECTION OF LIGHT. 

Molar Refraction in opaque substances. Translucent of metallic films. 

Ideal silver. Molecular structure; best polish obtainable. 

Reflectivity; definition; results of Fresnel, Potter, Jamin, 

and Conroy. Experimental method for reflectivities 

Polarizational analysis of reflected light ; experiments of Fresnel 

and Brewster; deviations from Fresnel's ^*-*'* Principal 



incidence and principal azimuth ; definition ; results for 
different metals by Jamin and Conroy . 

Fresnel's laws of reflection, and Green's theory, and observation 
graphically represented for water, glass, and diamond ; great 
deviations from Green's theory. Tangents of principal 
azimuth for seven transparent substances , 
Suggested correction of Green's theory ; velocity of condensational 
wave very small instead of very large ; by assuming negative 
compressibility ; voluminal instability and stability. Equa- 
tions of motion. Definition of wave-plane. Reflection and 
refraction at an interface between solids. Solutions for 
vibrations perpendicular to plane of incidence. Total internal 
reflection of an impulse and of sinusoidal waves ; forced 
clinging wave at interface in less dense medium. Differences 
of density and rigidity in the two mediums ; equal rigidities 
proved by observation ........ 

Solutions for vibrations in plane of incidence. Direct incidence. 

Rayleigh on the reflectivity of glass. Grazing incidence 
Opacity and reflectivity of metals ; metallic films ; electromagnetic 
screening. Continuity of dynamical wave-theory between 
light and slow vibrations of magnets to be discovered. Green's 
solution for vibrations in the plane of incidence realised ; case 
of infinite density . . ....... 

Quasi wave-motion in medium of negative or imaginary density ; 
ideal silver. Phase difference between components of reflected 
light. Principal incidence. Adamantine property 
Theory of Fresnel's rhomb. Phasal changes by internal reflection. 
Reflection of circularly polarized light Tables of phasal back- 
set by single reflection, and of virtual advance by two reflec- 
tions 



324-332 



332-343 



343-351 



363-373 



CONTENTS. 

Molar. Errors in construction of Fresnel's rhomb determined by 
MacCullagh ; errors of two Fre? iel's rhombs determined by 
observation. An eighty years 1 error regarding change of 
phase in total reflection ........ 

Orbital direction of light from a mirror. Rules for the orbital 
direction of light ; circularly or elliptically polarized light 
emerging from Fresnel's rhomb with illustrative diagram . 

Dynamics of loss of light in reflection ; principal azimuth is 45 if 
reflectivity perfect ......... 



XV 

PAGES 



393-403 

403-406 
406-407 



LECTURE XIX. 

RECONCILIATION BETWEEN FRESNEL AND GREEN. 

Molecular. Prof. Morley's numerical solutions of the problem of seven 

mutually interacting particles ...... 408-409 

Molar. Navier-Poisson doctrine disproved for ordinary matter by Stokes 
considering deviations in direction of greater resistance to 
compression, and by Thomson and Tait in the opposite direc- 
tion. Bulk-modulus of void ether practically infinite positive ; 
but negative just short of -n in space occupied by ponderable 
matter ; to reconcile Fresnel and Green. Velocity of conden- 
sational-rarefactional waves practically infinite in void ether, 
nearly zero in space occupied by ponderable matter ; return 
to hypothesis of 1887 with modification .... 409-412 

Molecular. Theory of interactions between atoms of ponderable matter and 

ether, giving dynamical foundation for these assumptions . 412-415 

Molar. Dynamical theory of adamantinism ; imaginary velocity of con- 
densational, rarefactional waves. Numerical reckoning of 
adamantinism defined (a). Adamantine reflection ; phasal 
lag ; diagrams of phasal lag for diamond. Principal incidence, 
and principal azimuth in adamantine reflection. Differences 
of principal incidence from Brewsterian angle very small ; 
values of the adamantinism of different substances . . 415-424 
Double refraction. Huyghens construction confirmed experimen- 
tally by Stokes. Fresnel's wave-surface verified in a biaxal 
crystal by Glazebrook. ^Eolotropic inertia, with the assump- 
tion of very small velocity for the condensational wave in 
transparent matter, gives dynamical explanation of Fresnel's 
laws of polarized light in a crystal. Archibald Smith's in- 
vestigation of Fresnel's wave-surface ..... 424-430 
Vibrational line proved not in wave-plane but perpendicular to 
ray by Glazebrook ; Glazebrook and Basset's construction for 
drawing the vibrational line ; calculation and diagram for 
arragonite. Theory wanted for vibrational amplitude in any 
part of wave-surface 430-435 



CONTENTS. 



LECTURE XX. 

Molecular Chiral rotation of the plane of polarization. Electro-etherial ex- 
planation of ieolotropic inertia. Dynamical explanation of 
chiral molecule, called a chiroid ... 

Molar. Formulas expressing chiral inertia in wave-motion ; gv 

Boussinesq in 1903. Imaginary terms and absorption ; a 
terms real; velocity of circularly polarized light in chiral 
medium Coexistent waves of contrary circular motions; 
sign of angular velocities about three mutually perpendicuh 
axes Fresnel's kinematics of coexistent waves of circular 
motions. Left-handed molecules make a right-handed medium 
Solution for wave-plane perpendicular to a principal axis. Optic 
chirality not imperceptible in a biaxal crystal ; experiment- 
ally determined in sugar by Dr H. C. Pocklington. Optic 
chirality in quartz crystal ; two trial assumptions cornered ; 
illustrative diagrams. Modification of double refraction by 

chirality in quartz 

Mathematical theories tested experimentally by McConnel. Ainc 
of rotation of vibrational line of jwlarized light travelling 
through various substances. Chirality in crystals and in 
crystalline molecules ; Sir George Stokes. Faraday's mag- 
neto-optic rotation . 

Molecular. Electro-etherial theory of the velocity of light through transparent 
bodies ; explanation of uniform rigidity ; theory of refraction. 
Energy of luminous waves in transparent substances 



PAGES 



436-440 



440-449 



450-468 



J.> -Mil 



461-467 



APPENDIX A. 
ON THE MOTION PRODUCED IN AN INFINITE ELASTIC SOLID BY 

THE MOTION THROUGH THE SPACE OCCUPIED BY IT OF A 
BODY ACTING ON IT ONLY BY ATTRACTION OR REPULSION. 

Suggestions regarding the distribution of ether within an atom, 

and regarding force between atoms and ether . 468-475 

Absolute orbits of ten particles of ether disturbed by a moving 
atom. Stream lines of relative motion of 21 particles of 
ether . 475-481 

Kinetic energy of the ether within a moving atom ; extra inertia 
of ether moving through a fixed atom. Velocity of light 
through an assemblage of atoms. Difficulty raised by Michel- 
son's experiment ; suggested explanation by Fitzgerald and 
by Lorentz of Leyden 481^185 



CONTENTS. XV11 

APPENDIX B, 

NINETEENTH CENTURY CLOUDS OVER THE DYNAMICAL 
THEORY OF HEAT AND LIGHT. 

PAGES 

Cloud I. The motion of ponderable matter through ether, an 
elastic solid. Early views of Fresnel and Young. Hypo- 
thesis of atoms and ether occupying the same space ; uniform 
motion of atom unresisted if less than the velocity of light. 
Hypothesis sufficient to explain the production of light by 
vibrating atom, and consistent with aberration. Difficulty 
raised by Michelson's experiment. Cloud I not wholly dis- 
sipated 486-492 

Cloud II. Waterstonian-Maxwellian distribution of energies ; 
demonstration by Tait for hard Boscovich atoms. The 
Boltzmann-Maxwell doctrine of the partition of energy ; its 
application to the kinetic theory of gases ; ratio of thermal 
capacities ; Clausius' theorem and the B.M. doctrine ; the 
latter not reconcilable with truth regarding the specific heats 
of gases 493-504 

Dynamical test-cases for the B.M. doctrine ; reflections of ball 
from elliptic boundary ; kinematic construction for finding 
the paths within boundary of any shape ; scalene triangle 
with rounded corners ; deviations from B.M. doctrine . . 504-512 

Statistics of reflections from corrugated boundary ; marlin spike- 
shaped boundary ; chattering impacts of rotators on a fixed 
plane ; particle constrained to remain near a closed surface ; 
ball bounding from corrugated floor ; caged atom struck by 
others outside ; non-agreement with B.M. doctrine . . 512-524 

B.M. doctrine applied to the equilibrium of a tall column of gas 
under gravity. Boltzmann and Maxwell and Rayleigh on the 
difficulties regarding the application of the B.M. doctrine to 
actual gases. Simplest way to get rid of the difficulties is to 
abandon the doctrine, and so dissolve Cloud II . . . 524-527 

APPENDIX C. 

ON THE DISTURBANCE PRODUCED BY TWO PARTICULAR FORMS OF 
INITIAL DISPLACEMENT IN AN INFINITELY LONG MATERIAL 
SYSTEM FOR WHICH THE VELOCITY OF PERIODIC WAVES 
DEPENDS ON THE WAVE-LENGTH. 

Solution in finite terms for elastic rod. Solution by highly con- 
vergent definite integral for all times and places in Poisson 
and Cauchy's deep-water wave problem ; four examples of 
results calculated by quadratures ; finite solution for deep- 
water waves due to initial disturbance .... 528-531 



xv iii CONTENTS. 

APPENDIX D. 
ON THE CLUSTERING OF GRAVITATIONAL MATTER IN 

ANY PART OF THE UNIVERSE. 

PAGES 

Ether is gravitationless matter filling all space. Total amount 
of ponderable matter in the space known to astronomers. 
Total of bright and dark star-disc apparent areas. Vast 
sphere, of perfectly compressible liquid, or of uniformly 
distributed atoms, given at rest; the subsequent motion 
under gravity. Clustering of gravitational matter distri- 
buted throughout infinite space. Atomic origin of all 
things; Democritus, Epicurus, Lucretius . . . . 532-540 

APPENDIX E. 
AEPINUS ATOMIZED. 

Electrion ; definition. Assumptions regarding atoms of ponderable 
matter and electrions and their interactions ; neutralizing 
quantum. Resinous and vitreous, instead of positive and 
negative. Overlapping of two mono-electrionic atoms ; case 
(1) each atom initially neutralized ; case (2) one atom void, 
the other neutralized ; in each case, after separation, the larger 
atom is left void ; probably true explanation of frictional 
and separational electricity ; commerce of energies involved 541-550 

Stable equilibrium of several electrions in an atom. Exhaustion 
of energy in stable groups of electrions. Electrionic explana- 
tion of Faraday's electro-inductive capacity, of Becquerel's 
radio-activity, and of electric conductivity .... 550-559 

Electrionic explanation of pyro-electricity and piezo-electricity ; 
di-polar electric crystal. Discoveries of Haiiy and the 
brothers Curie. Octo-polar electric seolotropy explained in 
a homogeneous assemblage of pairs of atoms, and in a 
homogeneous assemblage of single atoms. Voigt's piezo- 
electricity of a cubic crystal 



APPENDIX F. 

Dynamical illustrations of the magnetic and the helicoidal 
rotatory effects of transparent bodies on polarized light. 
Mounting of two-period pendulum rotated ; problem of the 
motion solved ; solution examined ; vibrations in rotating 
long straight rod of elliptic cross-section .... 569-5; 

Interior melting of ice; James Thomson's physical theory of 
the plasticity of ice ; illustrated by Tyndall's experiments. 
Stratification of vesicular ice by pressure observed in 



CONTENTS. 

glaciers, and experimentally produced oy Tyndall ; explained 
by the physical theory. Forbes explains blue veins in 
glaciers by pressure, and the resultant plasticity of a glacier 
by the weakness of the vesicular white strata. Tyndall's 
vesicles produced in clear ice by focus of radiant heat ; all 
explained by the physical theory . . . . 

APPENDIX G. 
HYDROKINETIC SOLUTIONS AND OBSERVATIONS. 

Part I. On the motion of free solids through a liquid. Eulerian 
equations of the motion, lines of reference ; axes fixed rela- 
tively to the body ; lines of reference fixed in space more 
convenient ; isotropic heh'9oid (now called chiroid) . 

Part II. Equations of motion when liquid circulates through 
holes in movable body ........ 

Part III. The influence of frictionless wind on waves in friction- 
less water ; particular wind velocities ; maximum wind velocity 
for level surface of water to be stable 

Part IV. Waves and ripples. Minimum wave-velocity. Moving 
force of ripples chiefly cohesion. The least velocity of fric- 
tionless air that can raise ripples on quiescent frictionless 
water. Effect of viscosity ; Stokes on the work against vis- 
cosity required to maintain a wave. Scott Russell's Report 
on Waves ; capillary waves 

Part V. Waves under motive power of gravity and cohesion 
jointly without wind. Cohesive force of water. Fringe of 
ripples in front of vessel ; ship-waves abaft on each quarter ; 
influence of cohesion on short waves. An experimental deter- 
mination of minimum velocity of solid to produce ripples or 
waves in water 

APPENDIX H. 
ON THE MOLECULAR TACTICS OF A CRYSTAL. 

Homogeneous assemblage of bodies ; theorem of Bravais. Thirteen 
nearest neighbours in acute-angled homogeneous assemblage. 
Equilateral assemblage. Stereoscopic representation of cluster 
of thirteen globes in contact. Barlow's model of homogeneous 
assemblages. Homogeneous assemblage implies homogeneous 
orientations. Homogeneous assemblage of symmetrical 
doublets 

How to draw partitional boundary in three dimensions ; parti- 
tional boundary for homogeneous plane figures. Homogeneous 
assemblage of tetrakaidekahedronal cells. Stereoscopic view 
of equilateral tetrakaidekahedron. Equivoluminal cells with 
minimum partitional area 



xix 

PAGES 



577-583 



584-588 
588-590 

590-593 



593-598 



598-601 



602-609 



609-617 



xx CONTENTS. 

PAGES 

Different qualities on two parallel sides of a crystal ; oppositely 
electrified by same rubber? oppositely electrified by changes 
of temperature; and by lateral pressures. Homogeneous 
packing of molecules or of continuous solids, as bowls or 
spoons or forks; case of single contact. Orientation for 
closest packing in this case. Definition of " chiral." Scalene 
convex solids homogeneously assembled. Ratio of void to 
whole space in heaps of gravel, sand, and broken stones. 
Orientation of molecules for closest packing. Change of 
configuration of assemblage in stable equilibrium causes 
expansion. Osborne Reynolds on the dilatation of wet sand 
when pressed by the foot; seen when glass-plate is sub- 
stituted for foot. Assemblage of globular molecules (or of 
ellipsoids if all similarly oriented) is rigid and constant in 
bulk, provided each molecule remains in coherent contact 

with twelve neighbours 617-626 

Geometrical problem of each solid touched by eighteen neighbours. 
Stereoscopic view of tetrahedron of convex scalene solids. 
Suggestion for a twin-crystal. Miller's definition of twin- 
crystal. Twin-plane as defined by Miller, and as defined 
by Stokes and Rayleigh. Successive twin-planes regular or 
irregular. Madan's change of tactics by heat in chlorate of 
potash ; Rayleigh 's experiment. Stokes and Rayleigh show 
that the brilliant colours of chlorate of potash crystals are 
due to successive twinnings at regular intervals ; cause of 
twinnings and counter-twinnings at regular intervals undis- 
covered 626-637 

Ternary tactics in lateral and terminal faces of quartz. The 
triple anti-symmetry required for the piezo-electricity in 
quartz investigated by the brothers Curie. Optic and piezo- 
electric chirality in quartz and tourmaline ; Voigt and Riecke. 
Orientational macliug, and chiral macling, of quartz. Geo- 
metrical theory of chirality 637-642 

APPENDIX I. 
ON THE ELASTICITY OF A CRYSTAL ACCORDING TO BOSCOVICH. 

Crystalline molecule and chemical molecule ; crystalline molecule 
probably has a chiral configuration in some or all chiral 
crystals ; Marbach, Stokes ; certainly however many chemical 
molecules have chirality ; magnetic rotation has no chirality. 
Forcive required to keep a crystal homogeneously strained. 
Boscovich's theory. Only 15 coefficients for simplest Bosco- 
vichian crystal 643 _ 646 

Displacement ratios and their squares and products. Work 
against mutual Boscovichian forces. Elastic energy of strained 



CONTENTS. 

simple homogeneous assemblage. > y ivier-Poisson relation 
proved for simple assemblage of Boscovich atoms. In an 
equilateral assemblage, when each point experiences force 
only from its twelve nearest neighbours, the diagonal rigidity 
is half the facial rigidity, relatively to the principal cube 
Single assemblage in simple cubic order. Equilateral assemblage ; 
elasticity moduluses ; conditions for elastic isotropy ; forces 
between points for equilibrium. Scalene homogeneous as- 
semblage of points ; scalene tetrahedron in which perpen- 
diculars from the four corners to opposite faces intersect in 
one point ; scalene assemblage giving an incompressible solid 
according to Boscovich. Twenty-one coefficients for elasticity 
of double assemblage ........ 



PAGES 



646-655 



655-661 



APPENDIX J. 
MOLECULAR DYNAMICS OF A CRYSTAL. 

Work done in separating a homogeneous assemblage of molecules 
to infinite distances from one another. Dynamics of double 
homogeneous assemblage. Distances of neighbours : in single 
equilateral assemblage: in double assemblage. Scalene as- 
semblage in equilibrium if infinite. Work-curves for two 



Stabilities of monatomic and diatomic assemblages ; stability not 
secured by positive moduluses ; stability of infinite row of 
similar particles ; Boscovich's curve ; equilibrium in the neigh- 
bourhood of ends of a long row of atoms. Alternating varia- 
tions of density seven or eight nets from surface of crystal 



662-671 



671-680 



APPENDIX K. 
ON VARIATIONAL ELECTRIC AND MAGNETIC SCREENING. 

Imperfectly conducting material acts as screen, only against steady 
or slowly varying electrostatic force. Blackened paper trans- 
parent for Hertz waves of high frequency; but opaque to 
light waves. Electro-conductive plate acts as screen against 
magnetic changes. Rotating conductor screens steady mag- 
netic force. Experimental illustration ..... 



681-687 



APPENDIX L. 

ELECTRIC WAVES AND VIBRATIONS IN A SUBMARINE 
TELEGRAPH WIRE, 



ERRATA. 

Page 14, line 19, for " refracted " read "reflected." 
15, delete footnote. 

39, line 7 from foot, for " vibrations " read " vibration." 

,, 46, line 7 from foot, for " V" read "p." 

48, in denominator of last term of equation (18), for "x* " read " r*." 

70, in heading, for " Part I." read "Part II." 

,, 73, line 5, before "the " insert " the reciprocals of the squares of." 

74, lines 2 and 6, for "e " read " t." 

75, in footnote, for "Appendix C " insert "Lect. XIX., pp. 408, 409," and 

delete "with illustrative curves." 

86, line 10 from foot, for " dx* " read " dt*." 

87, line 10, delete "that." 

87, line 3 from foot, for ll y-<f>- \ <t>" read "y J 0= 0." 

,, 88, line 15, for "words painting " read " word-painting." 

90 and 92, in heading, for " Part I." read " Part II." 

103, last line, for " I " read " 1 " . 

106, line 14, for "X" read " x ." 

239, for marginal " molecular " read " molar." 



/ 



NOTES OF LECTURES 

ON 

MOLECULAR DYNAMICS 

AND 

THE WAVE THEORY OF LIGHT. 

DELIVERED AT THE JOHNS HOPKINS UNIVERSITY, BALTIMORE, 



SIR WILLIAM THOMSON, 

PROFESSOR IN THE UNIVERSITY OF GLASGOW. 

STENOGRAPHIC ALLY REPORTED BY 
A. S. HATHAWAY, 

LATELY FELLOW IN MATHEMATICS OF THE JOHNS HOPKINS UNIVERSITY. 



1884. 

COPYRIGHT BY THE JOHNS HOPKINS UNIVERSITY, 
BALTIMORE, MD. 



ADVERTISEMENT. 



IN the month of October, 1884, Sir William Thomson of 
Glasgow, at the request of the Trustees of the Johns Hopkins 
University in Baltimore, delivered a course of twenty lectures 
before a company of physicists, many of whom were teachers of 
this subject in other institutions. As the lectures were not written 
out in advance and as there was no immediate prospect that they 
would be published in the ordinary form of a book, arrangements 
were made, with the concurrence of the lecturer, for taking down 
what he said by short-hand. 

Sir William Thomson returned to Glasgow as soon as these 
lectures were concluded, and has since sent from time to time 
additional notes which have been added to those which were 
taken when he spoke. It is to be regretted that under these 
circumstances he has had no opportunity to revise the reports. 
In fact, he will see for the first time simultaneously with the 
public this repetition of thoughts and opinions which were freely 
expressed in familiar conference with his class. The "papyro- 
graph" process which for the sake of economy has been employed 
in the reproduction of the lectures does not readily admit of cor- 
rections, and some obvious slips, such as Canchy for Cauchy, have 
been allowed to pass without emendation ; but the stenographer 
has given particular attention to mathematical formulas, and 
he believes that the work now submitted to the public may be 
accepted, on the whole, as an accurate report of what the lecturer 
said. 

A. S. HATHAWAY. 

Dec., 1884. 



LECTURE I. 
5 P:M. WEDNESDAY, Oct. 1, 1884. 



THE most important branch of physics which at present makes 
demands upon molecular dynamics seems to me to be the wave 
theory of light. When I say this, I do not forget the one great 
branch of physics which at present is reduced to molecular dyna- 
mics, the kinetic theory of gases. In saying that the wave theory 
of light seems to be that branch of physics which is most in want, 
which most imperatively demands, applications, of molecular dyna- 
mics just now, I mean that, while the kinetic theory of gases is a 
part of molecular dynamics, is founded upon molecular dynamics, 
works wholly within molecular dynamics, to it molecular dynamics 
is everything, and it can be advanced solely by molecular dynamics ; 
the wave theory of light is only beginning to demand imperatively 
applications of that kind of dynamical science. 

The dynamics of the wave theory of light began very molecu- 
larly in the hands of Fresnel, was continued so by Cauchy, and to 
some degree, though much less so, in the hands of Green. It was 
wholly molecular dynamics, but of an imperfect kind in the hands 
of Fresnel. Cauchy attempted to found his mathematical investi- 
gations on a rigorous molecular treatment of the subject. Green 
almost wholly shook off the molecular treatment, and worked out 
all that was to be worked out for the wave theory of light, by the 
dynamics of continuous matter. Indeed, I do not know that it is 
possible to add substantially to what Green has done in this sub- 
ject. Substantial additions are scarcely to be made to a thing 
perfect and circumscribed as Green's work is, on the explanation 
of the propagation of light, of the refraction and the reflection of 
light at the bounding surface of two different mediums, and of the 



R LECTURE I. 

O 

propagation of light through crystals, by a strict mathematical 
treatment founded on the consideration of homogeneous elastic 
matter Green's treatment is really complete in this respect, and 
there is nothing essential to be added to it. But there is a grea 
deal of exposition wanting to let us make it our own. We mus 
study it; we must try to see what there is in the very concise and 
sharp treatment, with some very long formulas, which we 
Green's papers. 

The wave theory of light, treated on the assumption tl 
medium through which the light is propagated is continuous and 
homogeneous, except where distinctly separated by a boundn 
inter-face between two different mediums, is really completed by 
Green. But there is a great deal to be learned from that kind of 
treatment that perhaps scarcely has yet been learned, because the 
subject has not been much studied nor reduced to a very popular 
form hitherto. 

Cauchy seemed unable to help beginning with the considera- 
tion of discreet particles mutually acting upon one another. 
But, except in his theory of dispersion, he virtually came to the 
same thing somewhat soon in his treatment every time he began 
it afresh, as if he had commenced right away with the considera- 
tion of a homogeneous, elastic solid. Green preceded him, I believe, 
in this subject. I have read a statement of Lord Rayleigh that there 
seems to have been an attributing to Cauchy of that which Green 
had actually done before. Green had exhausted the subject ; but 
there is I believe no doubt that Cauchy worked in a wholly inde- 
pendent way. 

What I propose in this first Lecture we must have a little 
mathematics, and I therefore must not be too long with any kind 
of preliminary remarks is to call your attention to the outstand- 
ing difficulties. The first difficulty that meets us in the dynamics 
of light is the explanation of dispersion; that is to say, of the fact 
that the velocity of propagation of light is different for different 
wave lengths or for light of different periods, in one and the same 
medium. Treat it as we will, vary the fundamental suppositions 
as much as we can, as much as the very fundamental idea allows 
us to vary them, and we cannot force from the dynamics of a 
homogeneous elastic solid a difference of velocity of wave propa- 
gation for different periods. 

Cauchy pointed out that if the sphere of action of individual 



INTRODUCTORY. 7 

molecules be comparable with the wave lengths, the fact of the 
difference of velocities for different periods or for different wave 
lengths in the same medium is explained. The best way, perhaps, 
of putting Cauchy's fundamental explanation is to say that there is 
heterogeneousness through space comparable with the wave length 
in the medium, that is, if we are to explain dispersion by 
Cauchy's unmodified supposition. We shall consider that, a little 
later. I have no doubt the truth is perfectly familiar already to 
many of you that it is essentially insufficient to explain the facts. 

Another idea for explaining dispersion has come forward more 
recently, and that is the assumption of molecules loading the 
luminiferous ether and somehow or other elastically connected with 
it. The first distinct statement that I have seen of this view is in 
Helmholtz's little paper on anomalous dispersion. I shall have 
occasion to speak of that a good deal and to mention other names 
whom Helmholtz quotes in this respect, so that I shall say 
nothing about it historically, except that there we have in Helm- 
holtz's paper, and by some German mathematicians who preceded 
him, quite another departure in respect to the explanation of 
dispersion. The Cauchy hypothesis gives us something comparable 
with the wave length in the geometrical dimensions of the body. 
Or, to take a crude matter of fact view of it, let us say the ratio of 
the distance from molecule to molecule (from the centre of one 
molecule to the centre of the next nearest molecule) to the wave 
length of light is the fundamental characteristic, as it were, to 
which we must look for the explanation of dispersion upon 
Cauchy's theory. 

We may take this fundamental idea in connection with the two 
hypotheses for accounting for dispersion: that we must have, in the 
very essence of the ponderable medium, some relation either to 
wave length or to period, and it seems at first sight (although this 
is a proposition that may require modification) that with very long 
waves the velocity of propagation should be independent of the 
period or wave length. That, at all events, seems to be the case 
when the subject is only looked upon according to Cauchy's view. 
We are led to say then that it seems that for very long waves 
there should be a constant velocity of propagation. Experiment 
and observation now seem to be falling in very distinctly to affirm 
the conclusions that follow from the second hypothesis that I 
alluded to to account for dispersion. In this second hypothesis, 



g LECTURE I. 

instead of having a geometrical dimension in the ponderable 
matter which is comparable with the wave length, we have a 
fundamental time-relation a certain definite interval of time 
somehow ingrained in the constitution of the ponderable matter, 
which is comparable with the period. So that, instead of a relation 
of length to length, we have a relation of time to time. 

Now, how are we to get our time element ingrained in the 
constitution of matter? We need scarcely put that question 
now-a-days. We are all familiar with the time of vibration of the 
sodium atom, and the great wonders revealed by the spectroscope 
are all full of indications showing a relation to absolute intervals of 
time in the properties of matter. This is now so well understood, 
that it is no new idea to propose to adopt as our unit of time one 
of the fundamental periods for instance, the period of vibration of 
light in one or other of the sodium D lines. You all have a 
dynamical idea of this already. You all know something about 
the time of vibration of a molecule, and how, if the time of 
vibration of light passing through any substance is nearly the 
same as the natural time of vibration of the molecules of the 
substance, this approximate coincidence gives rise to absorption. 
We all know of course, according to this idea, the old dynamical 
explanation, first proposed by Stokes, of the dark lines of the solar 
spectrum. 

We have now this interesting point to consider, that if we 
would work out the idea of dispersion at all, we must look definitely 
to times of vibration, in connection with all ponderable matter. To 
get a firm hypothesis that will allow us to work on the subject, let 
us imagine space, otherwise full of the luiniuiferous ether, to be 
partially occupied by something different from the general lumini- 
ferous ether. That something might be a portion of denser ether, 
or a portion of more rigid ether ; or we might suppose a portion of 
ether to have greater density and greater rigidity, or different 
density and different rigidity from the surrounding ether. We will 
come back to that subject in connection with the explanation 
of the blue sky, and, particularly, Lord Rayleigh's dynamics of the 
blue sky. In the meantime, I want to give something that will 
allow us to bring out a very crude mechanical model of dis- 
persion. 

In the first place, we must not listen to any suggestion that we 
are to look upon the luminiterous ether as an ideal way of putting 



INTRODUCTORY. 9 

the thing. A real matter between us and the remotest stars 
I believe there is, and that light consists of real motions of that 
matter, motions just such as are described by Fresnel and 
Young, motions in the way of transverse vibrations. If I knew 
what the electro-magnetic theory of light is, I might be able 
to think of it in relation to the fundamental principles of the wave 
theory of light. But it seems to me that it is rather a backward 
step from an absolutely definite mechanical notion that is put 
before us by Fresnel and his followers to take up the so-called 
Electro-magnetic theory of light in the way it has been taken up 
by several writers' of late. In passing, I may say that the one 
thing about it that seems intelligible to me, I do not think is 
admissible. What I mean is, that there should be an electric dis- 
placement perpendicular to the line of propagation and a magnetic 
disturbance perpendicular to both. It seems to me that when we 
have an electro-magnetic theory of light, we shall see electric dis- 
placement as in the direction of propagation, and simple vibrations 
as described by Fresnel with lines of vibration perpendicular to the 
line of propagation, for the motion actually constituting light. I 
merely say this in passing, as perhaps some apology is necessary 
for my insisting upon the plain matter-of-fact dynamics and the 
true elastic solid as giving what seems to me the only tenable 
foundation for the wave theory of light in the present state of our 
knowledge. 

The luminiferous ether we must imagine to be a substance 
which so far as luminiferous vibrations are concerned moves as if 
it were an elastic solid. I do not say it is an elastic solid. 
That it moves as if it were an elastic solid in respect to the lumini- 
ferous vibrations, is the fundamental assumption of the wave theory 
of light. 

An initial difficulty that might be considered insuperable is, 
how can we have an elastic solid, with a certain degree of rigidity 
pervading all space, and the earth moving through it at the 
rate the earth moves around the sun, and the sun and solar system 
moving through it at the rate in which they move through space, 
at all events relatively to the other stars ? 

That difficulty does not seem to me so very insuperable, 
Suppose you take a piece of Burgundy pitch, or Trinidad pitch, or 
what I know best for this particular subject, Scottish shoemaker's 
wax. This is the substance I used in the illustration I intend 



10 LECTURE I. 

to refer to. I do not know how far the others would succeed 
in the experiment. Suppose you take one of these substances, the 
shoemaker's wax, for instance. It is brittle, but you could form 
it into the shape of a tuning fork and make it vibrate. Take a 
long rod of it, and you can make it vibrate as if it were a piece of 
glass. But leave it lying upon its side and it will flatten down 
gradually. The weight of a letter, sealed with sealing-wax in the 
old-fashioned way, used to flatten it, will flatten it. Experiments 
have not been made as to the absolute fluidity or non-fluidity of 
such a substance as shoemaker's wax ; but that time is all that is 
necessary to allow it to yield absolutely as a fluid, is not an im- 
probable supposition with reference to any one of the substances 
I have mentioned. Scottish shoemaker's wax I have used in this 
way : I took a large slab of it, perhaps a couple of inches thick, 
and placed it in a glass jar ten or twelve inches in diameter. I filled 
the glass jar with water and laid the slab of wax in it with a 
quantity of corks underneath and two or three lead bullets on the 
upper side. This was at the beginning of an Academic year. Six 
months passed away and the lead bullets had all disappeared, and 
I suppose the corks were half way through. Before the year had 
passed, on looking at the slab I found that the corks were floating 
in the water at the top, and the bullets of lead were tumbled about 
on the bottom of the jar. 

Now, if a piece of cork, in virtue of the greater specific gravity 
of the shoemaker's wax would float upwards through that solid 
material and a piece of lead, in virtue of its greater specific gravity 
would move downwards through the same material, though only at 
the rate of an inch per six months, we have an illustration, it seems 
to me, quite sufficient to do away with the fundamental difficulty 
from the wave theory of light. Let the luminiferous ether be 
looked upon as a wax which is elastic and I was going to say 
brittle, (we will think of that yet of what the meaning of brittle 
would be) and capable of executing vibrations like a tuning fork 
when times and forces are suitable when the times in which the 
forces tending to produce distortion act, are very small indeed, and 
the forces are not too great to produce rupture. When the forces 
are long continued then very small forces suffice to produce un- 
limited change of shape. Whether infinitesimally small forces 
produce unlimited change of shape or not we do not know; but 
very small forces suffice to do so. All we need with respect to the 



INTRODUCTORY. 11 

luminiferous ether is that the exceedingly small forces brought 
into play in the luminiferous vibrations do not, in the times during 
which they act, suffice to produce any transgression of limits of 
distortional elasticity. The come-and-go effects taking place in 
the period of the luminiferous vibrations do not give rise to the 
consumption of any large amount of energy: not large enough an 
amount to cause the light to be wholly absorbed in say its pro- 
pagation from the remotest visible star to the earth. 

If we have time, we shall try a little later to think of some of 
the magnitudes concerned, and think of, in the first place, the mag- 
nitude of the shearing force in luminiferous vibrations of some 
assumed amplitude, on the one hand, and the magnitude of the 
shearing force concerned, when the earth, say, moves through the 
luminiferous ether, on the other hand. The subject has not been 
gone into very fully; so that we do not know at this moment 
whether the earth moves dragging the luminiferous ether altogether 
with it, or whether it moves more nearly as if it were through 
a frictionless fluid. It is conceivable that it is not impossible that 
the earth moves through the luminiferous ether almost as if it were 
moving through a frictionless fluid and yet that the luminiferous 
ether has the rigidity necessary for the performance of the lumini- 
ferous vibration in periods of from the four hundred million 
millionth of a second to the eight hundred million millionth of 
a second corresponding to the visible rays, or from the periods 
which we now know in the low rays of radiant heat as recently 
experimented on and measured for wave length by Abney and 
Langley, to the high ultra-violet rays of light, known chiefly by 
their chemical actions. If we consider the exceeding srnallness of 
the period from the 100 million millionth of a second to the 1600 
million millionth of a second through the known range of radiant 
heat and light, we need not fully despair of understanding the 
property of the luminiferous ether. It is no greater mystery at 
all events than the shoemaker's wax or Burgundy pitch. That is 
a mystery, as all matter is ; the luminiferous ether is no greater 
mystery. 

We know the luminiferous ether better than we know any 
other kind of matter in some particulars. We know it for its 
elasticity ; we know it in respect to the constancy of the velocity 
of propagation of light for different periods. Take the eclipses of 
Jupiter's satellites or something far more telling yet, the waxiugs 



12 LECTURE I. 

and wanings of self-luminous stars as referred to by Prof. Newcomb 
in a recent discussion at Montreal on the subject of the velocity of 
propagation of light in the lurainiferous ether. These phenomena 
prove to us with tremendously searching test, to an excessively 
minute degree of accuracy, the constancy of the velocity of propa- 
gation of all the rays of visible light through the luminiferous 
ether. 

Luminiferous ether must be a substance of most extreme 
simplicity. We might imagine it to be a material whose ultimate 
property is to be incompressible ; to have a definite rigidity for 
vibrations in times less than a certain limit, and yet to have the 
absolutely yielding character that we recognize in wax-like bodies 
when the force is continued for a sufficient time. 

It seems to me that we must come to know a great deal more 
of the luminiferous ether than we now know. But instead of 
beginning with saying that we know nothing about it, I say that 
we know more about it than we know about air or water, glass or 
iron it is far simpler, there is far less to know. That is to say, 
the natural history of the luminiferous ether is an infinitely 
simpler subject than the natural history of any other body. It 
seems probable that the molecular theory of matter may be so far 
advanced sometime or other that we can understand an excessively 
fine-grained structure and understand the luminiferous ether as 
differing from glass and water and metals in being very much more 
finely grained in its structure. We must not attempt, however, to 
jump too far in the inquiry, but take it as it is, and take the great 
facts of the wave theory of light as giving us strong foundations 
for our convictions as to the luminiferous ether. 

To think now of ponderable matter, imagine for a moment 
that we make a rude mechanical model. Let 
this be an infinitely rigid spherical shell ; let 
there be another absolutely rigid shell inside 
oi it, and so on, as many as you please. Na- 
turally, we might think of something more 
continuous than that, but I only wish to call 
your attention to a crude mechanical ex- 
planation, possibly sufficient to account for dispersion. Suppose 
we had luminiferous ether outside, and that this hollow space is 
of very small diameter in comparison with the wave length. Let 
zig-zag springs connect the outer rigid boundary with boundary 





INTRODUCTORY. 13 

number two. I use a zig-zag, not a spiral spring possessing the 
helical properties for which we are not ready yet, but which must 
be invoked to account for such properties as sugar and quartz 
have in disturbing the luminiferous vibrations. 
Suppose we have shells 2 and 3 also connected 
by a sufficient number of zig-zag springs and 
so on ; and let there be a solid nucleus in the 
centre with spring connections between it and 
the shell outside of it. If there is only one of 
these interior shells, you will have one definite 
period of vibration. Suppose you take away everything except 
that one interior shell ; displace that shell and let it vibrate while 
you hold the outer sheath fixed. The period of the vibration is 
perfectly definite. If you have an immense number of such 
sheaths, with movable molecules inside of them distributed through 
some portion of the luminiferous ether, you will put it into a 
condition in which the velocity of the propagation of the wave 
will be different from what it is in the homogeneous luminiferous 
ether. You have what is called for, viz., a definite period ; and 
the relation between the period of vibration in the light con- 
sidered, and the period of the free vibration of the molecule will 
be fundamental in respect to the attempt of a mechanism of that 
kind to represent the phenomena of dispersion. 

If you take away everything except one not too massive 
interior nucleus, connected by springs with the outer sheath, you 
will have a crude model, as it were, of what Helmholtz makes the 
subject of his paper on anomalous dispersion. Helmholtz, besides 
that, supposes a certain degree or coefficient of viscous resistance 
against the vibration of the nucleus, relatively to the sheath. 

If we had only dispersion to deal with there would be no 
difficulty in getting a full explanation by putting this not in a 
rude mechanical model form, but in a form which would commend 
itself to our judgment as presenting the actual mode of action of 
the particles, of gross matter, whatever they may be upon the 
luminiferous ether. It is difficult to imagine the conditions of the 
luminiferous ether in dense fluids or liquids, and in solids; but 
oxygen, hydrogen, and gases generally, must, in their detached 
particles, somehow or other act on the luminiferous ether, have 
some sort of elastic connection with it; and I cannot imagine 
anything that commends itself to our ideas better than this 



14 LECTURE I. 

multiple molecule which I have put before you. By taking 
enough of these interior shells, and by passing to the idea of 
continuous variation from the density of the ether to the enor- 
mously greater density of the molecule of grosser matter imbedded 
in it, we may come as it were to the kind of mutual action that 
exists between any particular atom and the luminiferous ether. 
It seems to me that there must be something in this molecular 
hypothesis, and that as a mechanical symbol, it is certainly not 
a mere hypothesis, but a reality. 

But alas for the difficulties of the undulatory theory of light ; 
refraction and reflection at plane surfaces worked out by Green 
differ in the most irreduceable way from the facts. They cor- 
respond in some degree to the facts, but there are differences that 
we have no way of explaining. A great many hypotheses have 
been presented, but none of them seems at all tenable. 

First of all is the question, are the vibrations of light per- 
pendicular to, or are they in, the plane of polarization defining 
the plane of polarization as the plane through the incident and 
refracted rays, for light polarized by reflection * Think of light 
polarized by reflection at a plane surface and the question is, are 
the vibrations in the reflected ray perpendicular to the plane of 
incidence and reflection, or are they in the plane of incidence and 
reflection ? I merely speak of this subject in the way of index. 
We shall consider very fully, Green's theory and Lord Rayleigh's 
work upon it, and come to the conclusion with .absolute certainty, 
it seems to me, that the vibrations must be perpendicular to the 
plane of incidence and reflection of light polarized by reflection. 

Now there is this difficulty outstanding the dynamical theory 
of refraction and reflection which gives this result does not give it 
rigorously, but only approximately. We have by no means so 
good an approach in the theory to complete extinction of the 
vibrations in the reflected ray (when we have the light in the 
incident ray vibrating in the plane of incidence and reflection) as 
observation gives. I shall say no more about that difficulty, be- 
cause it will occupy us a good deal later on, except to say that the 
theoretical explanation of reflection and refraction is not satisfac- 
tory. It is not complete; and it is unsatisfactory in this, that we 
do not see any way of amending it. 

But suppose for a moment that it might be mended and there 
is a question connected with it which is this : Is the difference 



INTRODUCTORY. 15 

between two mediums a difference corresponding to difference of 
rigidity, or does it correspond to difference of density ? That is an 
interesting question, and some of the work that had been done 
upon it seemed most tempting in respect to the supposition that 
the difference between two mediums is a difference of rigidity and 
not a difference of density. When fully examined, however, the 
seemingly plausible way of explaining the facts of refraction and 
reflection by difference of rigidity and no difference of density I 
found to be delusive, and we are forced to the view that there is 
difference of density and very little difference of rigidity. 

In working out this subject very carefully, and endeavouring to 
understand Lord Rayleigh's work upon it, and to learn what had 
been done by others, for a time I thought it too much of an 
assumption that the rigidity was exactly the same and that the 
whole effect was due to difference of density. Might it not be, it 
seemed to me, that the luminiferous ether on the two sides of the 
interface at which the refraction and reflection takes place, might 
differ both in rigidity and in density. It seemed to me then by a 
piece of work (which I must verify, however, before I can speak 
quite confidently about it *) that by supposing the luminiferous 
ether in the commonly called denser medium to be considerably 
denser than it would be were the rigidities equal, and the rigidity 
to be greater in it than in the other medium, we might get a 
better explanation of the polarization by reflection than Green's 
result gives. Green's work ends with the supposition of equal 
rigidities and unequal densities. He puts the whole problem in 
his formulae to begin with, but he ends with this supposition and 
his result depends upon it. 

Not to deal in generalities, let us take the case of glass and a 
vacuum, say. It seemed to me that by supposing the effective 
rigidity of luminiferous ether in glass to be greater than in vacuum 
and the density to be greater, but greater in a greater proportion 
than the rigidity, so that the velocity of propagation is less in glass 
than in vacuum, we should get a better explanation of the details 
of polarization by reflection than Green's result gives. 

It is only since I have left the other side of the Atlantic that I 
have worked at this thing, and going into it with keen interest, 
I inquired of everybody I met whether there were any observa- 
tions that would help me. At last I was told that Prof. Rood had 
* See Appendix A. 



16 LECTURE I. 

done what I desired to know, and on looking at his paper, I found 
that it settled the matter. 

My question was this: Has there been any measurement of the 
intensity of light reflected at nearly normal incidence from glass or 
water showing it to be considerably greater than Fresnel's formula 

gives? Fresnel gives ( -y) f r tne rat i f tne intensity of the 

reflected ray to the intensity of the incident ray in the case of 
normal incidence, or incidence nearly normal. I wanted to find 
out whether that had been verified by observation. It seems that 
nobody had done it at all until Prof. Rood, of Columbia College, 
New York, took it up. His experiments showed to a rather 
minute degree of accuracy an agreement with Fresnel's formula, 
so that the explanation I was inclined to make was disproved by 
it. I myself had worked with the reflection of a candle from a 
piece of window glass, during a pleasant visit to Dr Henry Field 
in your beautiful Berkshire hills at the end of August ; and had 
come to the same conclusion, even through such very crude and 
roughly approximate measurements. At all events, I satisfied 
myself that there was not so great a deviation from Fresnel's law 
as would allow me to explain the difficulties of refraction and 
reflection by assuming greater rigidity, for example, in glass than 
in air. We are now forced very much to the conclusion from 
several results, but directly from Prof. Rood's photometrical ex- 
periments, that the rigidity must be very nearly equal in the two. 

There is quite another supposition that might be made that 
would give us the same law, for the case of normal incidence ; the 
supposition that the reflection depends wholly upon difference 
of rigidity and that the densities are equal in the two. That gives 
rise to the same intensity of perpendicularly reflected light, so that 
the photometric measurement does not discriminate between these 
two extremes, but it does prevent us from pushing in on the other 
side of generally accepted result, a supposition of equal rigidities, 
in the manner that I had thought of. 

We may look upon the explanation of polarization by reflection 
and refraction as not altogether unsatisfactory, although not quite 
satisfactory : and you may see [pointing to diagrams which had 
been chalked on the board: see pp. 12 and 13 above] that this 
kind of modification of the luminiferous ether is just what would 
give us the virtually greater density. How this gives us precisely 



INTRODUCTORY. 17 

the same effect as a greater density I shall show when we work 
the thing out mathematically. We shall see that this supposition 
is equivalent to giving the luminiferous ether a greater density, 
while making the addition to virtual density according to the period 
of the vibration. 

I am approaching an end : I had hoped to get to it sooner. 
We have the subject of double refraction in crystals, and here 
is the great hopeless difficulty. 

If we look into the matter of the distortion of the elastic solid, Molar. 
we may consider, possibly, that that is not wonderful ; but 
Fresnel's supposition as to the direction of the vibration of light, is 
that the conclusion that the plane of vibration is perpendicular to 
the plane of polarization proves, if it is true, that the velocity 
of propagation of light in uniaxial crystals depends simply on 
the direction of vibration and not otherwise on the plane of 
the distortion. In the vibrations of light, we have to consider 
the medium as being distorted and tending to recover its shape. 
Let this be a piece of uniaxial crystal, Iceland spar, for instance, 
a round or square column, with its length in the direction of 
the optic axis, which I will represent on the board by a 
dotted line. 

Now the relation between light polarized by passing 
through Iceland spar on the one hand and light polarized 
by reflection on the other hand, shows us that if the line 
of vibration is perpendicular to the plane of polarization, the 
velocity of propagation of light in different directions through 
Iceland spar must depend solely on the line of vibration and not 
on the plane of distortion. 

There is no way in which that can be explained by the rigidity Case 
of an elastic solid. Look upon it in this way. Take a cube o 




Iceland spar, keeping the same direction of the axis direction 

* of axis: 

I vibration 
in plane < 
diagram. 



as before. Let the light be passing downwards, as 
indicated by the arrow-head. What would be the 
mode of vibration, with such a direction of propaga- 
tion ? Let us suppose, in the first place, the vibra- F - 2 
tions to be in the plane of the diagram. Then the dis- 
tortion of that portion of matter will be of the kind, and in the 
direction indicated by Fig. 2. A portion which was rectangular 
swings into the shape represented by the dotted lines. The force 
tending to cause a piece of matter which has been so displaced to 
T. L. 2 




18 LECTURE I. 

Molar. resume its original shape depends on this kind of distortion. The 
mathematical expression of it would be n a constant of rigidity, 
multiplied into a, the amount of the distortion. How that is to be 
reckoned is familiar to many of you, and we will not enter into the 
details just now. But just consider this other case, where the 
direction of propagation of the light is horizontal, as indicated by 
the arrow-head, that is to say, propagated perpen- 
dicular to the axis of the crystal (Fig. 3). What 
would be the nature of the distortion here, the 
vibration being still in the plane of the diagram ? 
The distortion will be in this way in which I move my 
two hands. A portion which was rectangular will swing into this 
shape, indicated by the dotted lines in Fig. 3. The return force 
will then depend upon a distortion of this kind. But a dis- 
tortion of this kind is identical with a distortion of that kind 
(Fig. 2), and the result must be, if the effect depends upon the 
return force in an elastic solid, that we must have the same velocity 
of propagation in this case and in that case (Figs. 3 and 2). 
Case III. Consider similarly the distortions produced by waves con- 
sisting of vibrations perpendicular to the plane of the diagrams, 
the arrow-heads in Figs. 2 and 3 still representing directions of 
propagation. The distortion in the case of Fig. 2 will still be 
as by shearing perpendicular to the axis, but in the direction 
perpendicular to the plane of the diagram, instead of that repre- 
sented by aid of the dotted lines of Fig. 2. Hence on the homo- 
geneous elastic solid theory, and in the case of an axially isotropic 
crystal, the velocity of propagation in this case would be the same 

* *' M in each f the first two cases- But witt vibration8 



Pro a 

tion per- dicular to the plane of the diagram, and propagation perpendicular 



the axis > the distortion is by shearing in the plane perpendicular 
vibrations to the line of the arrow in Fig. 3, and in the direction perpen- 
dicSar to dicular to tn ^ plane of the diagram. In the supposed crystal, the 
r ! gidity modulus f or this shearing would be different from the 
rigidity modulus for the shearings of cases I., II. and III. Hence 
cases III. and IV. would correspond to the extraordinary ray, and 
cases I. and II. the ordinary ray. 

Now observe, that light polarized in a plane through the axis, 
constitutes the ordinary ray, and light polarized in the plane per- 
pendicular to the plane of the ray and the axis constitutes the 
extraordinary ray. There is therefore an outstanding difficulty in 



INTRODUCTORY. 19 

the assumption that the vibration is perpendicular to the plane of Molar, 
polarization, which is absolutely inexplicable on the bare theory 
of an elastic solid. 

The question now occurs, may we not explain it by loading the 
elastic solid ? But the difficulty is, to load it unequally in different 
directions. Lord Rayleigh thought that he had got an explanation 
of it in his paper to which I have referred. He was not aware 
that Rankine had ^ad exactly the same idea. Lord Rayleigh at 
the end of this paper puts forward the supposition that difference of 
effective inertias in different directions may be adduced to explain 
the difference of velocity of propagation in Iceland spar. But 
if that were the case the wave propagation would not follow 
Huyghens' law. It would follow the law according to which 
the velocity of propagation would be inversely proportional to what 
it is according to Huyghens' law. Huyghens' geometrical construc- 
tion for the extraordinary ray in Iceland spar gives us an oblate 
ellipsoid of revolution according to which the velocity of propagation 
of light will be found by drawing from the centre of the ellipsoid 
a perpendicular to the tangent plane. For 
example (Fig. 4), CN will correspond to the 
velocity of propagation of the light when the 
front is in the direction of this (tangent) line. 
If the velocity is different in different directions Flg- 4< 

in virtue of an effective inertia, Lord Rayleigh's idea is that Moleculs 
the vibrating molecules might be like oblate spheroids vibrating in 
a frictionless fluid. The medium would thus have greater effective 
inertia when vibrating in the direction of its axis (perpendicular to 
its flat side), and less effective inertia when vibrating in its equa- 
torial plane. That is a very beautiful idea, and we shall badly want 
it to explain the difficulty. We would be delighted and satisfied 
with it if the pushing forward of the conclusions from it were veri- 
fied by experiment. Stokes has made the experiment. Rankine 
made the first suggestion in the matter, but did not push the ques- 
tion further than to give it as a mode of getting over the difficulty 
in double refraction. Stokes took away the possibility of it. He ex- 
perimented on the refracting index of Iceland spar for a variety of 
incidences, and found with minute accuracy indeed that Huyghens' 
construction was verified and that therefore it was impossible to 
account for the unequal velocity of propagation in different direc- 
tions by the beautiful suggestions of Rankine and Lord Rayleigh. 

22 




20 LECTURE I. 

Molecular. I have not been able to make a persistent suggestion of explan- 
ation, but I have great hopes that these spring arrangements are 
going to help us out of the difficulty. I will, just in conclusion, 
give you the idea of how it might conceivably do so. 

We can easily suppose these spring arrangements to have 
different strengths in different directions; and their directional 
law will suit exactly. It will give the fundamental thing we want, 
which is that the velocity of propagation of light shall depend on 
the direction of vibration, and not merely on the plane of the 
distortion. And it will obviously do this in such a manner as to 
verify Huyghens' law giving us exactly the same shape of wave- 
surface as the aeolotropic elastic solid would give. 

But alas, alas, we have one difficulty which seems still insuper- 
able and prevents my putting this forward as the explanation. It 
is that I cannot get the requisite difference of propagation al 
velocity for different directions of the vibration to suit different 
periods. If we take this theory, we should have, instead of the very 
nearly equal difference of refractive index for the different periods 
in such a body as Iceland spar, with dispersion merely a small 
thing in comparison with those differences : we should, I say, have 
difference of refractive index in different directions comparable 
with dispersion and modified by dispersion to a prodigious degree, 
and in fact we should have anomalous dispersion coming in between 
the velocity of propagation in one direction and the velocity of 
propagation in another. The impossibility of getting a difference 
of wave velocity in different directions sufficiently constant for the 
different periods seems to me at present a stopper. 

So now, I have given you one hour and seven minutes and 
brought you face to face with a difficulty which I will not say 
is insuperable, but something for which nothing ever has been 
done from the beginning of the world to the present* time that 
can give us the slightest promise of explanation. 

I shall do to-morrow, what I had hoped to do to-day, give you 
a little mathematics, knowing that it is not going to explain every- 
thing. But I think that in relation to the wave-theory of light we 
really have an interest in working out the motions of an elastic 
solid and obtaining a few solutions that depend on the equations of 
motion of an elastic solid. I shall first take the case of zero 

* [I retract this statement. See an Appendix near the end of the present 
volume. W. T. Oct. 28, 1889.] 



INTRODUCTORY. 21 

rigidity ; that will give us sound. We shall take the most elemen- Molecular 
tary sounds possible, namely a spherical body alternately expand- 
ing and contracting. We shall pass from that to the case of a 
single globe vibrating to and fro in air. We shall pass from that 
to the case of a tuning-fork, and endeavour to explain the cone of 
silence which you all know in the neighbourhood of a vibrating 
tuning-fork. I hope we shall be able to get through that in a 
short time and pass on our way to the corresponding solutions 
of the motions of a wave proceeding from a centre in respect to the 
wave-theory of light. 



LECTURE II. 
Oct. 2, 5 P.M. 



PART I. In the first place, I will take up the equations of 
motion of an elastic solid. I assume that the fundamental prin- 
ciples are familiar to you. At the same time, I should be very 
glad if any person present would, without the slightest hesitation, 
ask for explanations if anything is not understood. I want to be 
on a conferent footing with you, so that the work shall be rather 
something between you and me, than something in which I shall 
be making a performance before you in a matter in which many 
of you may be quite as competent as I am, if not more so. 

I want, if we can get something done in half an hour, on these 
problems of molar dynamics as we may call it, to distinguish from 
Molecular dynamics, to come among you, and talk with you for 
a few moments, and take a little rest; and then go 011 to a problem 
of molecular dynamics to prepare the way for motions depending 
on mutual interference among particles under varying circum- 
stances that may perhaps have applications in physical science and 
particularly to the theory of light. 

The fundamental equations of equilibrium of elastic solids are, 
of course, included in D'Alembert's form of the equations of 
motion. I shall keep to the notation that is employed in Thomson 
and Tait's Natural Philosophy, which is substantially the same 
notation as is employed by other writers. 

Let a, b, c denote components of distortion : a, is a distortion in 
the plane perpendicular to OX produced by slippings parallel to 
either or to both of the two co-ordinate planes which intersect 
in OX 

Let us consider this state of strain, in which, without other 
change, a portion of the solid in the plane yz which was a 



DYNAMICS OF ELASTIC SOLID. 23 

square becomes a rhombic figure. The measurement of that state Molar, 
of strain is given fully in Thomson and Tait's geometrical pre- 
liminary for the theory of elastic solids, (Natural Philosophy, 
169177, and 669, or Elements, 148156 and 640). It 
is called a simple shear. It may be measured 
either by the rate of shifting of parallel planes per 
unit distance perpendicular to them, or, which 
comes to exactly the same thing, the change of 
the angle measured in radians. Thus I shall write 
down inside this small angular space the letter a, 
to denote the magnitude of the angle, measured 
in radians; the particular case of strain considered 
being a slipping parallel to the plane YOX. 

I use the word " radian "; it is not, hitherto, a very common 
word; but I suppose you know what I mean. In Cambridge 
in the olden time we used to have a very illogical nomen- 
clature, "the unit angle" a very absurd use of the definite 
article "the". It is illogical to talk of an angle being measured in 
terms of " the " unit angle ; there is no such thing as measuring 
anything, except in terms of "a" unit. The degree is a unit 
angle; so is the minute; so is the second; so is the quadrant; 
so is the round ; all of these are units in frequent use for angular 
measurement. The unit in which it is most generally convenient 
to measure angles in Analytical Mechanics is the angle whose arc 
is radius. That used to be called at Cambridge " the unit angle." 
My brother, James Thomson, proposed to call it " the radian', 

There are three principal distortions, a, b, c, relative to the 
axes of OX, OY, OZ\ and again, three principal dilatations con- 
densations of course if any one is negative, e, f, g, which are 
the ratios of the augmentation of length to the length. 

The general equation of energy will of course be an equation in 
which we have a quadratic function of e, f, g, a, b, c, the ex- 
pression for which will be 
(Ile 2 + 12e/+ 13eg + 14ea + loeb + IGec + 21e/+ 22/ 2 + 23/7 + ) 

We do not here use 11, 12,... etc., as numbers but as symbols 
representing the twenty-one coefficients of this quadratic subject 
to the conditions 12 = 21, etc. If we denote this quadratic function 
by E, then 

d = lie + 12/+ 130 + 14a + 156 + 16c. 



24 



LECTURE II. PART I. 



Molar. This is a component of the normal force required to produce this 
compound strain e, f, g, a, b, c. According to the notation of 
Thomson and Tait, let 



dE 



, R 



dE 
dg> 



T 

; . -i 

da 



dE 
db' 



dc' 



We have then, the relation 



the well-known dynamical interpretation of which you are of course 
familiar with. A little later we shall consider these 21 coefficients, 
first, in respect to the relations among them which must be 
imposed to produce a certain kind of symmetry relative to the 
three rectangular axes ; and then see what further conditions must 
be imposed to fit the elastic solid for performing the functions 
of the luminiferous ether in a crystal. 

Before going on to that we shall take the case of a perfectly 
isotropic material. We can perhaps best put it down in tabular 
form in this way : 



a 


ft) 


& 











u 


a 


to 











Li 


& 


a 




















n 














n 




















n 



In the first place in this lower right-hand corner-square which 
has to do with the distortions a, b, c, alone, if we let n represent the 
rigidity-modulus the three main diagonal terms will each be n, and 
those not in the diagonal will be zero. Six of the 21 coefficients 
are thus determined as follows : 

44 = 55 = 66 = n, and 45 = 4G = 56 = 0, 
and other nine of them, by the zeros in the upper right and lower 

left corner-squares, 

14 = 15 = 16 = 24 = 25-26 = 34 = 35 = 36 = 0. 



DYNAMICS OF ELASTIC SOLID. 25 

To verify these zeros of the upper right-hand and lower left-hand Molar, 
corner-squares, let us consider what possible relations there can be 
for an isotropic body between longitudinal strains and distortions. 
Clearly none. No one of the longitudinal strains can call into 
play a tangential force in any of the faces ; and conversely, if the 
medium be isotropic, no distortion produced by slipping in the 
faces parallel to the principal planes can introduce a longitudinal 
stress a stress parallel to any of the lines OX, OY, OZ. There- 
fore we have all zeros in these two squares. We know that 
11 = 22 = 33; and each of these will be represented by Saxon A ($&). 
Now consider the , effect of a longitudinal pull in the direction 
of OX. If the body be only allowed to yield longitudinally, that 
clearly will give rise to a negative pull in the directions parallel to 
Oy, Oz. We have then a cross connection between pulls in the 
directions OX, OF, OZ. Isotropy requires that the several 
mutual relations be all equal, so that we have just one coefficient 
to express these relations. That coefficient is denoted by Saxon B 
(23). Thus we fill up our 36 squares, which represent but 21 co- 
efficients in virtue of the relations 12 = 21, etc. We can now 
write down our quadratic expression for the energy, 

E = K&O 2 +/ 2 + f) + 223 (fg + ge + ef) + n (a 2 + 6 2 + c 2 ;]. 

Instead of these Saxon letters &, 23, which have very distinct and 
obvious interpretations, we may introduce the resistance of the 
solid to compression, the reciprocal of what is commonly called 
the compressibility, or, what we may call the bulk modulus, k. 
Then it is proved in Thomson and Tait, and in an article in the 
Encyclopedia Britannica which perhaps some of you may have*, 
that 1 = k + 1 n, 23 = k f n. The considerations which show 
these relations with the bulk modulus also show us that we must 
have n = J(& 23). This is most important. Take a solid cube 
with its edges parallel to OX, Y, OZ. Apply a pull along two 
faces perpendicular to OX and an equal pressure on two faces 
perpendicular to OF; that will give a distortion in the plane xy. 
Find the value of that simple shear ; it is done in a moment. Find 
the shearing force required to produce it calculated from & and 23, 
and equate that to the force calculated from the rigidity modulus 
n, and then you find this relation. The relations for complete 

* Beprint of Mathematical and Physical Papers, Vol. in. Art. xcn. now in the 
Press. [W. T. Aug. 7, 1885.] 



2(3 LECTURE II. PART I. 

isotropy are exhibited here in this quadratic expression for the 
energy, if in it we take $ (& - 13) in place of n. 

We shall pass on to the formation of the equations of motion. 
For equilibrium, the component parallel to OX of the force applied 
at any point x, y, z of the solid, reckoned per unit of bulk at that 
point must be equal to 

(dP + dU + dT\ 



if the body be held distorted in any way, by bodily forces applied 
all through the interior ; because the resultant of the elastic force 
on an infinitesimal portion of matter at the point x, y, z is obviously 



+ + 3 dxdydz. To prove this remark that if the pull 

augments as you go forward in the direction OX there will, in 
virtue of that, be a resultant forward pull 

Y u dP 

-jdx.dydz upon the infinitesimal ele- 

^u ment. The two tangential forces, U per- 
pendicular to OF, and U perpendicular 
to OX on the one pair of forces and the 
pair of forces equal and in opposite di- 
rections on the other faces constitute two 
balancing couples, as it were. If this 
tangential force parallel to OX increases 

as we proceed in the direction y positive, there will result a positive 
force on the element, because it is pulled to left by the smaller 
and to right by the larger, and thus the force in the direction 

of OX receives a contribution , dy.dzdx. Quite similarly we 

7/rr 

find, -7- dz . dxdy as a third and last contribution to the force 
parallel to OX. 

Now, let there be no bodily forces acting through the material, 

but let the inertia of the moving part, and the reaction against 

acceleration in virtue of inertia, constitute the equilibrating re- 

action against elasticity. The result is, that we have the equation 

dP dU dT_ tff 

dx + dy + dz~ P df 

if by p we denote the density and by we denote the displace- 
ment from equilibrium in the direction OX of that portion of 
matter having x, y, z for co-ordinates of its mean position. 



DYNAMICS OF ELASTIC SOLID. 27 

I said I would use the notation of Thomson and Tait who Molar. 
employ a, #, 7 to denote the displacements; but errors are too 
common when a and a are mixed up, especially in print, so we 
will take , ij, instead. I have had trouble in reading Helm- 
holtz's paper on anomalous dispersion, on this account, very fre- 
quently not being able to distinguish with a magnifying glass 
whether a certain letter was a or a. 

The values of S, T, U, we had better write out in full, although 
the others may be obtained from the value of any one of the 
three by symmetry. The expenditure of chalk is often a saving 
of brains. They are : 



We have P= 'j&e + i3 (/+#) There are two or three other forms 
which are convenient in some cases, and I will put them down 
(writing in for k + ^ri) 

. d . (di) d\ , (d% d<n d\ , dt 

P = (m + n) - + (m - n) ( - + = (m - ri) l-f + ~ + - ) + 2n ~ 

J dx ' \dy dzj '\dx dy dz) dx 



- -j- - -- -~-- 

\dx dy dzj \dx dy dz 

We shall denote very frequently by 8 the expression 

d + dr, + dS 

dx dy dz ' 
so that for example, the second of these expressions is 

P = (m - n) B + 2n ^ . 

If we want to write down the equations of a heterogeneous medium, 
as will sometimes be the case, especially in following Lord Ray- 
leigh's work on the blue sky, we must in taking dP/dx, dP/dy, &c., 
to find the accelerations, keep the symbols m, n inside of the 
symbols of differentiation ; but for homogeneous solids, we treat m 
and n as constant. I forgot to say that 8 is the cubic dilatation or 
the augmentation of volume per unit volume in the neighbour- 
hood of the point x, y, z, which is pretty well known, and helps us 
to see the relations to compressibility. If we suppose zero rigidity, 
P = mS is the relation between pressure and volume. In order 
to verify this take the preceding expression for P and make n = 
and we obtain P = mS, the equation for the compression of a 
compressible fluid, in which m has become the bulk modulus. 



28 LECTURE II. PART II. 

Molar. This sort of work I have called molar dynamics. It is the 

dynamics of continuous matter; there are in it no molecules, no 
heterogeneousnesses at all. We are preparing the way for dealing 
with heterogeneousnesses in the most analytical manner by sup- 
posing ra and n to be functions of x, y, z. Lord Rayleigh 
studied the blue sky in that way, and very beautifully ; his treat- 
ment is quite perfect of its kind. He considers an imbedded 
particle of water, or dust, or unknown material, whatever it is 
that causes the blue sky. To discover the effect of such a particle, 
on the waves of light, he supposes a change of rigidity and of 
density from place to place in the luminiferous ether; not an 
absolutely sudden change, but confined to a space which is small 
in comparison with the wave-length. 

Ten minutes interval. 

Molecular. PART II. I want to take up another subject which will pre- 
pare the way to what we shall be doing afterward, which is the 
particular dynamical problem of the movement of a system of con- 
nected particles. I suppose most of you know the linear equations 
of motion of a connected system whose integral always leads to 
the same formula as the cycloidal pendulum ; this result being 
come to through a determinant equated to zero, giving an alge- 
braical equation whose roots are essentially real for the square 
of the period of any one of the fundamental modes of simple 
harmonic motion. 

As an example, take three weights, one of 7 pounds, another of 
14 pounds, and another of 28 pounds, say. The lowest weight is 
hung upon the middle weight by a spiral spring ; 
the middle is hung upon the upper by a spiral 
spring, and the upper is attached to a fixed point 
by a spiral (or a zig-gag) spring. It is a pretty 
illustration; and I find it very useful to myself. 
I am speaking, so to say, to professors who sym- 
pathize with me, and might like to know an 
experiment which will be instructive to their 
pupils. 

Just apply your finger to any one of the 
weights, the upper weight, for example. You 
soon learn to find by trial the fundamental periods. Move it up 
and down gently in the period which you find to be that of the 




INTRODUCTION TO PART II. 29 

three all moving in the same direction. You will get a very Moleculi 
pretty oscillation, the lowest weight moving through the greatest 
amplitude, the second through a less, and the upper weight through 
the smallest. That is No. 1 motion, corresponding to the greatest 
root of the cubic equation which expresses the solution of the 
mathematical problem. No. 2 motion will come after a little 
practice. You soon learn to give an oscillation a good deal quicker 
than before, the first; a second mode, in which the lowest weight 
moves downward while the two upper move upwards, or the two 
lower move downwards while the upper moves upwards, or it 
might be that the middle weight does not move at all in this 
second mode, in which case the excitation must be by putting the 
finger on the upper or the lower weight, These periods depend 
upon the magnitude of the weights, and the strength of the 
springs that we use, and are soon learned in any particular set of 
weights and springs. It might be a good problem for -junior 
laboratory students to find weights and springs which will in- 
sure a case of the nodal point lying between the upper and 
middle weights, or at the middle weight, or between the middle 
and lower weights. The third mode of vibration, corresponding 
to the smallest root of the cubic equation, is one in which you 
always have one node in the spring between the upper and middle 
weights, and another node in the spring between the middle and 
lowest (the first and third weight vibrating in the same direction, and 
the middle weight in an opposite direction to the first and third). 
It is assumed that there is no mass in the springs. If you 
want to vary your laboratory exercises, take smaller masses for the 
weights, and more massive springs, and if you omit the attached 
masses altogether, you pass on to a very beautiful illustration 
of the velocity of sound. For that purpose a long spiral spring 
of steel wire, the spiral 20 feet long, hung up, say, if you have 
a lofty enough room, will answer, and you will readily get 
two or three of the graver fundamental modes without any 
attached weights at all. In the special problem which we have 
been considering we have three separate weights and not a con- 
tinuous spring ; and we have three, and only three modes of vibra- 
tion, when the springs are massless. We have an infinite number 
of modes when the mass of the springs is taken into account. In 
any convenient arrangement of heavy weights, the stiffness of the 
springs is so great and their masses so small that the gravest 



30 LECTURE II. PART II. 

Molecular, period of vibration of one of the springs by itself will be very short ; 
but take a long spring, a spiral of best pianoforte steel wire, if you 
please, and hang it up, with a weight perhaps equal to its own, 
on its lower end, and you will find it a nice illustration for getting 
several of the graver of the infinite number of the fundamental 
modes of the system. 

I want to put down the dynamics of our problem for any 
number of masses. You will see at once that that is just the case 
that I spoke of yesterday, of extending Helmholtz's singly vibrat- 
ing particle connected with the luminiferous ether to a multiple 
vibrating heavy elastic atom imbedded in the luminiferous ether, 
which I think must be the true state of the case. A solid mass 
must act relatively to the luminiferous ether as an elastic body 
imbedded in it, of enormous mass compared with the mass of the 
luminiferous ether that it displaces. In order that the vibrations 
of the- ether may not be absolutely stopped by the mass, there 
must be an elastic connection. It is easier to say what must 
be than to say that we can understand how it comes to be. The 
result is almost infinitely difficult to understand in the case of 
ether in glass or water or carbon disulphide, but the luminiferous 
ether in air is very easily imagined. Just think of the molecules 
of oxygen and nitrogen as if each were a group of ponderable par- 
ticles mutually connected by springs, and imbedded in homogene- 
ous perfectly elastic jelly constituting the luminiferous ether. 
You do not need to take into account the gaseous motions of the 
particles of oxygen, nitrogen, and carbon dioxide in our atmosphere 
when you are investigating the propagation of luminous waves 
through the air. Think of it in this way : the period of vibration 
in ultra-violet rays in luminous waves and in infra red heat 
waves so far as known, is from the 1600 million-millionth of 
a second to the 100 million-millionth of a second. Now think 
how far a particle of oxygen or nitrogen moves, according to the 
kinetic theory of gases, in the course of that exceedingly small 
time. You will find that it moves through an exceedingly small 
fraction of the wave-length. For example, think of a molecule 
moving at the rate of 50000 centimetres per second. In the period 
of orange light it crawls along lO' 10 of a centimetre which is only 
1/600000 of the wave-length of orange light. I am fully confident 
that the wave motion takes place independently of the translatory 
motion of the particles of oxygen and nitrogen in performing 



DYNAMICS OF A ROW OF CONNECTED PARTICLES. 



31 



their functions according to the kinetic theory of gases. You may Moleculai 
therefore really look upon the motion of light waves through our 
atmosphere as being solved by a dynamical problem such as this 
before us, applied to a case in which there is so little of effective 
inertia due to the imbedded molecules, that the velocity of light is 
not diminished more than about one-thirty-third per cent, by it. 
More difficulties surround the subject when you come to consider 
the propagation of light through highly condensed gases, or trans- 
parent liquids or solids. 

In our dynamical problem, let the masses of the bodies be 
represented by w,, ra. 2 ,...7W r I am going to suppose the several 
particles to be acted upon by connecting springs. I do not want 
to use spiral springs here. The helicalness of the spring in these 
experiments has no sensible effect; but we want to introduce 
a spiral for investigating the dynamics of the helical properties, as 
shown by sugar. It is usually called the rotatory property, but this 
is a misnomer. The magneto-optical property which was dis- 
covered by Faraday is rotational, the property exhibited by quartz 
and sugar and such things, has not the essential elements of 
rotation in it, but has the characteristic of a spiral spring (a helical 
spring, not a flat spiral), in the constitution of the matter that ex- 
hibits it. We apply the word helical to the one and the word 
rotational to the other. 

I am going to suppose one other connect- _ 

ed particle P, which is moved to and fro -^^ c 

with a given motion whose displacement 
downwards from a fixed point 0, we shall call 
f. Let c x be the coefficient of elasticity of 
the first spring, connecting the particle P 
with the particle m^, c 2 the coefficient of 
elasticity of the next spring connecting m 1 
and ra 2 ; c >+1 , the coefficient of elasticity of 
the spring connecting m, to a fixed point. 
We are not taking gravity into account ; we 
have nothing to do with it. Although in the 
experiment it is convenient to use gravity, 
it would be still better if we could go to the 
centre of the earth and there perform the 
experiment. The only difference would be, 
these springs would not be pulled out by the weights hung upon 




32 LECTURE II. CONCLUSION. 

Molecular, them. In all other respects the problem would be the same, and 
the same symbols would apply. 

We are reckoning displacements downwards as positive, tl 
displacement of the particle m t being a?,. The force acting upon 
m in virtue of the spring connection between it and P is c, ( - a;,) ; 
and in virtue of the spring connection between it and TO, is the 
opposing pull - c 2 (x, - ar t ) ; so that the equation of motion of the 
first particle is 



For No. 2 particle we have 

m, ^r' = C 2 (x.- *,) - c s (a, - a,) ; and so on. 
Now suppose P to be arbitrarily kept in simple harmonic mo- 
tion in time or period r ; so that f = const, x cos . We assume 

that every part of the apparatus is moving with a simple harmonic 
motion, as will be the case if there were infinitesimal viscous 
resistance and the simple harmonic motion of P is kept up long 

27T< 

enough ; so that we can write rr, = const, x cos ^ , etc. 

going to alter the ms so as to do away with the 47T* which comes 

in from differentiation. I will let ^ denote the mass of the first 

4*7T 

particle, and ^\ the mass of the second particle, etc. The result 

^TT 

will be that the equations of motion become 

~ ^ x i = c i (f - *i) - c ( x i ~ *) etc - 

Our problem is reduced now to one of algebra. There are 
some interesting considerations connected with the determinant 
which we shall obtain by elimination from these equations. To 
find the number of terms is easy enough ; and it will lead to some 
remarkable expressions. But I wish particularly to treat it with 
a view to obtaining by very short arithmetic the result which can 
be obtained from the determinant in the regular way only by 
enormous calculation. We shall obtain an approximation, to the 
accuracy of which there is no limit if you push it far enough, 



LECTURE II. CONCLUSION. 33 

that will be exceedingly convenient in performing the calcu- Molecular, 
lations. 

In the next lecture, resuming the molar problem, we shall 
begin with the solution for sound, of the equations that are now 
before you on the board. We shall next try to go on a step further 
with this molecular problem, of the vibrations of our compound 
molecule. 



T. L. 



LECTURE III. 
Oct. 3, SP.M. 



Molar. WE will now go on with the problem of Molar dynamics, the 
propagation of sound or of light, from a source. I advise you all 
who are engaged in teaching, or in thinking of these things for 
yourselves, to make little models. If you want to imagine the 
strains that were spoken of yesterday, get such a box as this 
covered with white paper and mark upon it the directions of the 
forces 8, T, U. I always take the directions of the axes in a 
certain order so that the direction of positive rotation shall be 
from y to z, from z to x, from x to y. What we call positive is the 
same direction as the revolution of a planet seen from the northern 
hemisphere, or opposite to the motion of the hands of a watch. I 
have got this box for another purpose, as a mechanical model of an 
elastic solid with 21 independent moduluses, the possibility of 
which used to be disproved, and after having been proved, was still 
disbelieved for a long time. 
Let us take our equations, 

d* dP dU dT 



=M 
dx \d dx 



A 
) 



We shall not suppose that ra and n are variables, but take them 
constant. If we do not take them constant we shall be ready for 
Lord Rayleigh's paper on the blue sky, already referred to. I will 
do the work upon the board in full, as it is a case in which 
the expenditure of chalk saves brain ; but it would be a waste to 



WAVES OR VIBRATIONS OF ELASTIC SOLID. 35 

print such calculations, for the reason that a reader of mathematics Molar. 
should always have pencil and paper beside him to work the thing 
out. * * * * The result is that 



We take the symbol 



. _ , , 
V -M+d d*' 



In the case of no rigidity, or n = 0, the last term goes out. We 
shall take solutions of these equations, irrespectively of the 
question of whether we are going to make n = or not, and we 
shall find that one standard solution for an elastic solid is in- 
dependent of n and is therefore a proper solution for an elastic 
fluid. 

I have in my hand a printed report* of a Royal Institution 
lecture of Feb. 1883, on the Size of Atoms, containing a note on 
some mathematical problems which I set when I was examiner for 
the Smith's Prizes at Cambridge, Jan. 30, 1883. One was to show 
that the equations of motion of an isotropic elastic solid are what 
we have here obtained, and another to show that so and so was a 
solution. We will just take that, which is : Show that every pos- 
sible solution of these three equations [(1) etc.] is included in the 
following : 

% = f + u, v=f- + v, S=f+w ............ (2), 

dx dy dz 

where <f>, u, v, w, are some functions of x, y, z, t\ with the condition 
that u, v, w are such that 

*.*>?_ ........................ 

dx dy dz 
If we calculate the value of the cubic dilatation, we find 

8.^ + *f+* + f!_^ ....... (4). 

dx dy dz 

Again, by using g = -r-+u in (1), we find (bearing in mind 



* Keprinted in Vol. I. of Sir W. Thomson's Popular Lectures and Addresses. 
(Macmillan, 1889). 

32 



LECTURE TIT. PART T. 

Now we may take 



C6) - 



. 

The full justification and explanation of this procedure is 
reserved. [See commencement of Lecture IV. below.] Multiply 
(6) by dx, and the corresponding y and z equations by dy, dz, and 
add. We thus get a complete differential ; in other words, the 
relation which <f> must satisfy is 



And (2) shows that if <J> satisfies (7) we have , v, w, satisfying 
equations of the same form, but with n instead of (m + n) ; viz. 



By solving these four similar equations, one involving (m + w), 
and three involving n, we can get solutions of (1), that is certain. 
That we get every possible solution, I shall hope to prove 
to-morrow. The velocity of the sound wave, or condensational 

. The velocity of the wave of distortion in the 

elastic solid is A/-. I shall not take this up because I am very 

anxious to get on with the molecular problem ; but you see 
brought out perfectly well the two modes of waves in an isotropic 
homogeneous solid, the condensational wave and the distortional 
wave. The condensational wave follows the equations of motion 
of sound, which is the same as if n were null ; and this gives the 
solution of the propagation of sound in a homogeneous medium, 
like air, etc. The solution is worked out ready to hand for the 
distortional wave because the same forms of equations give us 
separate components , v, w ; the same solution that gives us the 
velocity potential for the condensational waves, gives us the 
separate components of displacement for the distortional waves. 

What I am going to give you to-morrow will include a solution 
which is alluded to by Lord Rayleigh. There is nothing new in it. 
I am going to pass over the parts of the solution which interpreted 
by Stokes explain that beautiful and curious experiment of 
Leslie's. Lord Rayleigh quotes from Stokes, ending his quotation 
of eight pages with "The importance of the subject and the masterly 
manner in which it has been treated by Prof. Stokes will probably 



WAVES OR VIBRATIONS OF ELASTIC FLUID. 37 

be thought sufficient to justify this long quotation." I would just Molar. 
like to read two or three things in it. Lord Rayleigh says (Theory 
of Sound, Vol. ii. p. 207), "Prof. Stokes has applied this solution 
to the explanation of a remarkable experiment by Leslie, according 
to which it appeared that the sound of a bell vibrating in a 
partially exhausted receiver is diminished by the introduction 
of hydrogen. This paradoxical phenomenon has its origin in the 
augmented wave length due to the addition of hydrogen in conse- 
quence of which the bell loses its hold (so to speak) on the 
surrounding gas." I do not like the words " paradoxical pheno- 
menon;" "curious phenomenon," or "interesting phenomenon" 
would be better. There are no paradoxes in science. Lord 
Rayleigh goes on to say, " The general explanation cannot be 
better given than in the words of Prof. Stokes : ' Suppose a person 
to move his hand to and fro through a small space. The motion 
which is occasioned in the air is almost exactly the same as it 
would have been if the air had been an incompressible fluid. 
There is a mere local reciprocating motion in which the air 
immediately in front is pushed forward and that immediately 
behind impelled after the moving body, while in the anterior space 
generally the air recedes from the encroachment of the moving 
body, and in the posterior space generally flows in from all sides 
to supply the vacuum that tends to be created ; so that in lateral 
directions, the flow of the fluid is backwards, a portion of the 
excess of the fluid in front going to supply the deficiency 
behind.'" It will take some careful thought to follow it. I 
wish I had Green here to read a sentence of his. Green 
says, " I have no faith in speculations of this kind unless they 
can be reduced to regular analysis." Stokes speculates, but is 
not satisfied without reducing his speculation to regular analysis. 
He gives here some very elaborate calculations that are also 
important and interesting in themselves, partly in connection 
with spherical harmonics, and partly from their exceeding instruc- 
tiveness in respect to many problems regarding sound. Passing 
by all that five or six pages of mathematics I will not tax your 
brains with trying to understand the dynamics of it in the course 
of a few minutes ; I am rather calling your attention to a thing 
to be read than reading it Stokes conies more particularly to 
Leslie's experiments. Instead of a bell vibrating, Stokes con- 
siders the vibrations of a sphere becoming alternately prolate and 



38 



LECTURE HI. PAUT II. 



Molar. 



Molecular. 



oblate ; and he shows that the principles are the same. Read all 
this for yourselves. I have intended merely to arouse an mteres 
in the subject. 

Ten minutes interval. 

To return to the consideration of our molecules connected by 
springs, we will suppose a good fixing at the top, so firm and stiff 
that the changing pull of the spring does not give it any sensible 
motion. For any one of the springs let there be a certain change 
of pull, c per unit change of length. This coefficient c measures 
what I call the longitudinal rigidity of the spring : its effective 
stiffness in fact. It is not the slightest consequence whether the 
spring is long or short, only, if it is long, let it be so much the 
stiffer ; but long or short, thick or thin, it must be massless. I 
mean that it shall have no inertia. The masses may be equal 

,_ __, ur unequal, ami HIV coiim-ctcd l>v >pi'in--. L--t Ufl 

attach here something like the handle of the bell 
pull of pre-electric ages ; something that you can 
pull by. Call it P. This, in our application to the 
luminiferous ether, will be the rigid shell lining be- 
tween the luminiferous ether and the first moving mass. 
The equation of motion for the first mass be- 
comes, on bringing f to the left-hand side, 




and similarly for the second mass ; I shall use t 
to denote any integer. I find the letter i too useful 
for that purpose to give it up, and when I want 
to write the imaginary J 1, I use i. Let us call 
the first coefficient on the right a,, the similar 
coefficient in the next equation a s ,and so on, so that 



The ith equation will thus be 

- c<# 4 ., = ap t + C M X M . 

Now write down all these j equations supposing the whole number 
of the springs to be j; form the determinant by which you find all 
of the others in terms of f , and the problem is solved. 

If we had a little more time I would like to determine the 
number of terms in this determinant. We will come back to 



VIBRATIONS OF SERIAL MOLECULE. 39 

that because it is exceedingly interesting ; but I want at once to Molecuiai 
put the equations in an interesting form, taking a suggestion from 
Laplace's treatment of his celebrated Tidal problem. What we 
want is really the ratios of the displacements, and we shall there- 
fore write 



introducing the sign minus, so that when the displacements are 
alternately positive and negative the successive ratios will be all 
positive. We have then, 

c c * c 2 c, 2 

-ii- = u=a. a , u 9 = , s- , . . . u t = a, , . . . 
- x r w 2 w 3 w m 

w> = a ; ; (w, +1 = oo ). 

We can now form a continued fraction which, for the case that 
we want, is rapidly convergent. If this be differentiated with 
respect to r~ 2 , we find a very curious law, but I am afraid we 
must leave it for the present. The solution is 




Thus if we are given the spring connections and the masses, 
everything is known when the period is known. If you develop 
this, you simply form the determinant ; but the fractional form 
has the advantage that in the case when the masses are larger 
and larger, and the spring connections are not larger in pro- 
portion, we get an exceedingly rapid approximation to its value 
by taking the successive convergeuts. The differential coefficient 
of this continued fraction with respect to the period is essentially 
negative, and thus we are led beautifully from root to root, and 
see the following conditions : First, suppose we move P to and 
fro in simple harmonic motion of very short period ; then when 
the whole has got into periodic movement, it is necessary that P 
and the first particle move in opposite directions. The vibrations 
of the first particle needs to be " hurried up " (if you will allow 
me an expressive American phrase) when the motion of P is 
of a shorter period than the shortest of the possible inde- 
pendent motions of the system with P held fixed. Now if you 
want to hurry up a vibrating particle, you must at each end of 
its range press it inwards or towards its middle position. 



40 LECTURE III. PART IT. 

Molecular. You meet this principle quite often ; it is well known in 
the construction of clock escapements. To hurry up the vibra- 
tory motion of our system we must add to the return force of 
particle No. 1 by the action of the spring connected to the 
handle P, by moving P always in the direction opposite to the 
motion of m. From looking at the thing, and learning to under- 
stand it by feeling the experiment, if you do not understand it by 
brains alone, you will see that everything that I am saying is 
obvious. But it is not satisfactory to speak of these things in 
general terms unless we can submit them to a rigorous analysis. 

I now set the system in motion, managing, as you see, to get 
it into a state of simple harmonic vibration by my hand applied 
to P. That, which you now see, is a specimen of the configuration 
in which the motion of P is of a shorter period than the shortest 
of the independent motions with P fixed. Suppose now, the 
vibration of P to be less rapid and less rapid ; a state of things 
will come, in which, the period of P being longer and longer, the 
motion of the first particle will be greater and greater. That is 
to say, if I go on augmenting the period of P we shall find for 
the same range of motion of P, that the ranges of motion of m l 
and of the other particles generally will be greatly increased 
relatively to the range which I give to P. In analytical words, 
if we begin with a configuration of values corresponding to T 
very small, and then, if we increase T to a certain critical 

value, we shall find ' will become infinite. In the first place, we 

begin with u,, 2 ,... M, all positive; and r small enough will make 
them all positive as you see. Now take the differential coefficient 
of u t with respect to r and it will be found to be essentially nega- 
tive. In other words, if we increase T, we shall diminish u |f ti t .... 
In every case u, will first pass through zero and become negative. 

When u l is zero we have the first infinity -J = x . If we diminish 

T a little further u a will pass through zero to negative while u, is 
still negative. Diminish T a little further and u s will become 
zero and pass to negative, while u g is still negative; but in the 
mean time u l may have reached a negative maximum and passed 
through zero to positive, or it may not yet have done so. We 
shall go into this to-morrow ; but I should like to have you know 
beforehand what is going to come from this kind of treatment of 
the subject. 



LECTURE IV. 

Oct. 4, 3.45 P.M. 
We found yesterday Molar. 

^ n. . ,_\ dS . 2 ,^ 



and we saw that we get two solutions, which when fully inter- 
preted, correspond to two different velocities of propagation, on 
the assumptions that were put before you as to a condensational 
or a distort! onal wave. We will approach the subject again from 
the beginning, and you will see at once that the sum of these 
solutions expresses every possible solution. 

In one of our solutions of yesterday, we took, instead of f , 77, , 
other symbols u, v, w, which satisfied the condition, 
du dv dw _ 

~T "T" ~^t~ I ~~j V, 

ax ay dz 

In other words, the u, v, w of yesterday express the displacements 
in a case in which the dilatation or condensation is zero. Now, just 
try for the dilatation in any case whatever, without such restriction. 
This we can do as follows : Differentiate (1) with respect to x 
(taking account of the constancy of ra and ri) and the correspond- 
ing equations with respect to y and ^, and add. We thus find 



42 LECTURE IV. 

Molar. This equation, you will remember, is the same as we had yesterday 
for 6. We shall consider solutions of this equation presently ; 
but now remark, that whatever be the displacements, we have 
a dilatation corresponding to some solution of this equation. It 
may be zero, but it must satisfy (2). Now in any actual case, B is 
a determinate function of x, y, z, t\ and whether we know it or 
not, we may take 6 to denote a function such that 

y 2 </> = B through all space .................. (3). 

This function, 6, is determinate. It is in fact given explicitly 
(as is well known in the theory of attraction) by the equation 



where B' denotes the value of B at (&', y', z). This formula (4) is 
important as giving 6 explicitly ; and exceedingly interesting on 
account of the relations to the theory of attraction ; but in the 
wave-problem, when B is given. (3) gives 6 determinately and in 
the easiest possible way. Putting now 



(16 d6 

v > s = + 



(5). 



We have B = y 8 <f> + -.- + - + -j- ...(6); 

dx dy dz 

and therefore, by (3), T -- 4- -=- -f -=- = . 

dx dy dz "*'' 

Now, remembering that (1) are satisfied by dQjdx, d6/dt/, d6/dz 
in place of f, ij, f, we see, by multiplying the first by dx, or the 
second by dy, or the third by dz, and integrating, that 
dty = 

and we find the three equations for 

<- d6 d6 d6 

dx ' dy ' dz 

reduced, in virtue of (7), to the following : 

- M = 2 M -* v = n s d ' W - 

For any possible solutions of equations (1), we have a value 
of B which is a function of x, y, z; take the above volume integral 



WAVES OR VIBRATIONS IN ELASTIC SOLID. 43 

corresponding to this value of B through all points of space x'y ' z ', 
and we obtain the corresponding < function which fulfils the 

condition \7 2 < = B. Now, let us compound displacements ~ , etc., 
with the actual displacements and denote the resultant as follows: 

dd> dd> dd> 

s ~ j~ u > i} ~r =v > b ^r =w, 
dx dy dz 

and remarking that therefore 

du dv dw 

T- + T- + -7T = > 
dx dy dz 

we see the proposition that we had before us yesterday established. 
Hence to solve the three equations (1) we have simply to find 
B by solution of the one equation 



and to deduce <f> from B, by (3); or to find <f> direct by solution 
of the equation 



and u, v, w from the three separate similar equations with n in the 
place of (m + n), subject to the conditions 

du dv dw _ 
dx dy dz 

We shall take our <J> equation and see how we can from it 
obtain different forms of d> solutions. We can do that for the 
purpose of illustrating different problems in sound, and in order 
to familiarize you with the wave that may exist along with the 
wave of distortion in any true elastic solid which is not incom- 
pressible. We ignore this condensational wave in the theory of 
light. We are sure that its energy at all events, if it is not null, 
is very small in comparison with the energy of the luminiferous 
vibrations we are dealing with. But to say that it is absolutely 
null would be an assumption that we have no right to make. 
When we look through the little universe that we know, and think 
of the transmission of electrical force and of the transmission of 
magnetic force and of the transmission of light, we have no right 
to assume that there may not be something else that our philo- 



44 LECTURE IV. 

sophy does not dream of. We have no right to assume that there 
may not be condensational waves in the luminiferous ether. We 
only do know that any vibrations of this kind which are excited 
by the reflection and refraction of light are certainly of very small 
energy compared with the energy of the light from which they 
proceed. The fact of the case as regards reflection and refraction 
is this, that unless the luminiferous ether is absolutely incom- 
pressible, the reflection and refraction of light must generally 
give rise to waves of condensation. Waves of distortion may 
exist without waves of condensation, but waves of distortion cannot 
be reflected at the bounding surface between two mediums without 
exciting in each medium a wave of condensation. When we come 
to the subject of reflection and refraction, we shall see how to 
deal with these condensational waves and find how easy it is to 
get quit of them by supposing the medium to be incompressible. 
But it is always to be kept in mind as to be examined into, are 
there or are there not very small amounts of condensational waves 
generated in reflection and refraction, and may after all, the pro- 
pagation of electric force be by these waves of condensation ? 

Suppose that we have at any place in air, or in luminiferous 
ether (I cannot distinguish now between the two ideas) a body 
that through some action we need not describe, but which is con- 
ceivable, is alternately positively and negatively electrified; may 
it not be that this will give rise to condensational waves ? Suppose, 
for example, that we have two spherical conductors united by a 
fine wire, and that an alternating electromotive force is pro- 
duced in that fine wire, for instance by an "alternate current" 
dynamo-electric machine ; and suppose that sort of thing goes on 
away from all other disturbance at a great distance up in the 
air, for example. The result of the action of the dynamo-electric 
machine will be that one conductor will be alternately positively 
and negatively electrified, and the other conductor negatively and 
positively electrified. It is perfectly certain, if we turn the 
machine slowly, that in the air in the neighbourhood of the con- 
ductors we shall have alternately positively and negatively directed 
electric force with reversals of, for example, two or three hundred 
per second of time with a gradual transition from negative 
through zero to positive, and so on; and the same thing all 
through space; and we can tell exactly what the potential and 
what the electric force is at each instant at any point. Now, does 



ELECTRIC WAVES. 45 

any one believe that if that revolution was made fast enough Molar, 
the electro -static law of force, pure and simple, would apply to 
the air at different distances from each globe? Every one believes 
that if that process be conducted fast enough, several million times, 
or millions of million times per second, we should have large devia- 
tion from the electrostatic law in the distribution of electric force 
through the air in the neighbourhood. It seems absolutely certain 
that such an action as that going on would give rise to electrical 
waves. Now it does seem to me probable that those electrical 
waves are condensational waves in luminiferous ether ; and pro- 
bably it would be that the propagation of these waves would be 
enormously faster than the propagation of ordinary light waves. 

I am quite conscious, when speaking of this, of what has 
been done in the so-called Electro-Magnetic theory of light. 
I know the propagation of electric impulse along an insulated 
wire surrounded by gutta percha, which I worked out myself 
about the year 1854, and in which I found a velocity comparable 
with the velocity of light*. We then did not know the relation 
between electro-static and electro-magnetic units. If we work 
that out for the case of air instead of gutta percha, we get simply 
" v" (that is, the number of electrostatic units in the electro- 
magnetic unit of quantity,) for the velocity of propagation of the 
impulse. That is a very different case from this very rapidly 
varying electrification I have ideally put before you : and I have 
waited in vain to see how we can get any justification of the way 
of putting the idea of electric and magnetic waves in the so-called 
electro-magnetic theory of light. 

I may refer to a little article of mine in which I gave a sort 
of mechanical representation of electric, magnetic, and galvanic 
forces galvanic force I called it then, a very badly chosen name. 
It is published in the first volume of the reprint of my papers. 
It is shown in that paper that the static displacement of an elastic 
solid follows exactly the laws of the electro-static force, and that 
rotatory displacement of the medium follows exactly the laws of 
magnetic force. It seems to me that an incorporation of the 
theory of the propagation of electric and magnetic disturbances 
with the wave theory of light is most probably to be arrived at 
by trying to see clearly the view that I am now indicating. In 
the wave theory of light, however, we shall simply suppose the 
* (See an Appendix near the end of the present volume.) 



46 LECTURE IV. 

Molar, resistance to compression of the luminiferous ether and the 
velocity of propagation of the condensational wave in it to be 
infinite. We shall sometimes use the words "practically infinite" 
to guard against supposing these quantities to be absolutely 
infinite. 

I will now take two or three illustrations of this solution for 
condensational waves. Part of the problem that I referred to 
yesterday says : prove that the following is a solution of (7), the 
equation of motion, 



or, if we put for brevity, 



.(11). 



The question might be put into more analytical form : to find 
a solution of (7) isotropic in respect to the origin of co-ordinates ; 
or to solve (7) on the assumption that <f> is a function of r and t. 
Taking this then as our problem, remark that we now have (be- 
cause $ is a function of r) 



Hence (7), with both sides multiplied by ?- s , becomes 

p'r-^w*^ <"> 

of which the general solution is 



where ^/^ 

and F and f denote two arbitrary functions. 

This result simply expresses wave disturbance of r<f>, with 
velocity of propagation V[(fc + f n)/p] : and it proves (9), being 
merely the case of a simple harmonic wave disturbance propagated 
in the direction of r increasing, that is to say outwards from the 
origin. 



CONDENSATIONAL WAVES IN SOLID OR FLUID. 47 

Here then is the determination of a mode of motion which 
is possible for an elastic solid. We shall consider the nature of 

this motion presently. The factor - in (9) prevents it from being 

a pure wave motion. Passing over that consideration for the 
present, we note that it is less and less effective, relatively to the 
motion considered the farther we go from the centre. 

In the meantime, we remark that the velocity of propagation 
in an elastic solid is but little greater than in a fluid with the 
same resistance to compression, k is the bulk modulus and 
measures resistance to compression, n is the rigidity modulus. 
I may hereafter consider relations between k and n for real solids. 
k is generally several times n, so that %n is small in comparison 
with k, and therefore in ordinary solids the velocity of propagation 
of the condensational wave is not greatly greater than if the solid 
were deprived of rigidity and we had an elastic fluid of the same 
bulk modulus. 

I shall want to look at the motion in the neighbourhood of 
the source. That beautiful investigation of Stokes, quoted by 
Lord Rayleigh, has to do entirely with the region in which the 

change of value of the factor f-J from point to point is consider- 

able. Without looking at that now, let us find the components of 
the displacement and their resultant, and study carefully all the 
circumstances of the motion. 

-?>-> j > are the three components of the displacement. 

Clearly, therefore, the displacement will be in the direction of the 
radius because everything is symmetrical ; and its magnitude will 

be ^: and from (11) and (10) we find 
dr 

dd> -1 . 27J-1 

^ = __ sm? + __ 



Having obtained this solution of our equations, let us see what 
we can make of interpreting it. When r is great in comparison 

with 5- , the first term becomes very small in comparison with 



48 LECTURE IV. 

Molar, the second and we have 

j j .> i 

(16)*. 



Therefore, when the distance from the origin is a great many 
wave-lengths, the displacement is sensibly equal to - cos q, 

and is therefore approximately in the inverse proportion to the 
distance ; and the intensity of the sound if the solution were 
to be applied to sound, would be inversely as the square of the 
distance from the source. 

I want now to get a second and a third solution. Take 



as the velocity potential for a fresh solution. I take it that you 
all know that if we have one solution <, for the velocity potential, 
we can get another solution by ^ any linear function of 

64 djj> d<j> 

dx ' dy ' dz' 
Now let us find the displacements 

djr <ty <ty 

dx ' dy ' dz ' 

Here I want to prove that though this solution is no longer sym- 
metrical with respect to r, so that there will be motions other 
than radial in the neighbourhood of the source, yet still the 
motion is approximately radial at great distances from the source. 
Work it out, and you will find that 

ety 2ir sin? [V / X \' S - 3*>1 cos q r 8 - So? 
dx ~ \ r |y teJ ~V J + ~P " of 

The principal term here is - f sin q. We might go on to the 

A, i 

third and fourth and higher differential coefficients of <j>, with 
their larger and larger numbers of terms. The interpretation of 
this multiplicity of terms, of the terms other than those which 
I am now calling the " principal terms," is all-important in respect 
to the motion of the air in the neighbourhood of the source. It 

* I use = to denote approximate equality. 

t X/2rr is introduced merely for convenience. The solution differs from 
t = d<f>l<lx, only by a constant factor. 



AERIAL WAVES OF SOUND. 49 

is dealt with in that splendid work of Stokes, one of the finest Molar. 
things ever written in physical mathematics, of which I read to 
you this afternoon, with reference to the effect of an atmosphere 
of hydrogen round a bell killing its sound. But we will drop 
those terms and think only of the terms which express the 
efficiency of the vibrator at distances great in comparison with 
the wave-length. 

Thus, for the ^-component displacement of the motion now 
considered, we have 



This approximate equality is true for distances from the centre 
great in comparison with the wave-length. Let me remark, it is 
the differentiation of cos q that gives the distantly effective terms 
of the displacement ; and in differentiating ty with respect to y, 
you have simply to differentiate cos q and to take the differential 
coefficient of r now with respect to y, instead of x as formerly. 
So that we may write down the principal terms of the y and z 
displacements by taking y\x and zjx of the second member of (19) 
as follows : 

2-7T XV . 2-7T XZ . 



The three component displacements being proportional to x, y, z, 
shows that the resultant displacement is in the direction of the 

radius ; and its magnitude is - % s ^ n <? If we write x = r cos i, 
this becomes 

2-7T COS i . 

~T~r~ * .......................... ( )j 

or the displacement is inversely proportional to the distance. If 
i = we have a maximum ; if i = 5- we have zero. The upshot of 

it is that the displacement is a maximum in the axis OX, zero 
everywhere in the plane of OY, OZ; and symmetrical all round 
the axis OZ. 

A third solution is got by taking -y- as our velocity potential. 

At a distance from the origin, great in comparison with the wave- 
T. L. 4 



50 LECTURE IV. 

length, the displacement is in the direction of the radius, and its 

d tf$ 

magnitude is -r- -5-5 . 

Now the interpretation of these cases is as follows : The first 
solution, (velocity potential <) a globe alternately becoming larger 
and smaller; the second solution, (velocity potential dfyjdx) a 
globe vibrating to and fro in a straight line ; the third solution, 
(velocity potential d*^dsf) a characteristic constituent of the 
motion of the air produced by two globes vibrating to and fro 
in the line of their centres, or by the prongs of a vibrating fork. 

This last requires a little nice consideration, and we shall take 
it up in a subsequent lecture. The third mode does not quite 
represent the motion in the neighbourhood of the pair of vibrating 
globes, or of the prongs of a vibrating fork ; there must be an 
unknown amount of the first mxle compounded with the third 
mode for this purpose. The expression for the vibration in the 
neighbourhood of a tuning fork, going so far from the ends of it 
that \ve will be undisturbed or but little disturbed, by the general 
shape of the whole thing, will be given by a velocity potential 

A(f> + -^. That will be the velocity potential for the chief terms, 

the terms which alone have effect at great distances. The dif- 
ferentiation will be performed simply with reference to the r in 
the term sin q or cos q ; and will be the same as if the coefficient 
of sin q or cos q were constant. A differentiation of this velocity 
potential will show that the displacement is in the direction of 
the radius from the centre of the system, and the magnitude of 

the displacement will be jf : ( A$ + ^*f ) . 

A is an unknown quantity depending upon the tuning fork. 
I want to suggest this as a junior laboratory exercise, to try tuning 
forks with different breadths of prongs. When you take tuning 
forks with prongs a considerable distance asunder you have much 
less of the <J> in the solution : try a tuning fork with flat prongs, 
pretty close together, and you will find much more of the <f>. 
The < part of the velocity potential corresponds to the alternate 
swelling and shrinking of the air between the two prongs of the 
tuning fork. The larger and flatter the prongs are the greater is 
the proportion of the </> solution, that is to say, the larger is the 



AERIAL VIBRATIONS ROUND A TUNING FORK. 51 

value of A in that formula; and the smaller the angle of the cone 
of silence. 

The experiment that I suggest is this : take a vibrating tuning 
fork and turn it round until you find the cone of silence, or find 
the angle between the line joining the prongs and the line going to 
the place where your ear must be to hear no sound. The sudden- 
ness of transition from sound to no sound is startling. Having 
the tuning fork in the hand, turn it slowly round near one ear 
until you find its position of silence. Close the other ear with your 
hand. A very small angle of turning round the vertical axis from 
that position gives you a startlingly loud sound. I think it is very 
likely that the place of no sound will, with one and the same fork, 
depend on the range of vibration. If you excite it very powerfully, 
you may find less inclination between the line of vibration of the 
prongs and the line to the place of silence ; less powerfully, greater 
inclination. It will certainly be different with different tuning forks, 



4-2 



LECTURE V. 
SATURDAY, Oct. 4, 5 P.M. 

I STATED in the last lecture that the second solution, corre- 
sponding to the velocity potential -jz , would represent the effect, 

at a great distance from the mean position, of a single body vibrat- 
ing to and fro in a straight line. I said a sphere, but we may take 
a body of any shape vibrating to and fro in a straight line; and at 
a very great distance from the vibrator, the motion produced will 

be represented by the velocity potential -.- , provided the period 

of the vibration is great in comparison with the time taken by 
sound to travel a distance equal to the greatest diameter of the 

body. Then the velocity potential , *> in the third solution, 

(13C 

would, I believe, represent (without an additional term A<f>) the 
motion at great distances, when the origin of the sound consists 
in two globes, let us say, for fixing the ideas, placed at a distance 
from one another very great in comparison with their diameters 
and set to vibrate to and fro through a range small in comparison 
with the distance between them, but not necessarily small in 
comparison with their diameters : provided always that the period 
of the vibration is great in comparison with the time taken by 
sound to travel the distance between the vibrators. Suppose 
this is a globe in one hand, and this is one in the other. I now 
move my hands towards and from each other the motion of the 
air produced by that sort of motion of the exciting bodies would, at 
a very great distance, be expressed exactly by the velocity potential 



But when you have two globes, or two flat bodies, very near 
one another, you need an unknown amount of the < vibration to 



AERIAL VIBRATIONS ROUND A TUNING FORK. 53 

represent the actual state of the case. That unknown amount Molar: 
might be determined theoretically for the case of two spheres. i a tion! U 
The problem is analogous to Poisson's problem of the distribution 
of electricity upon two spheres, and it has been solved by Stokes 
for the case of fluid motion (see Mem. de I'lnst., Paris, 1811, pp. 1, 
163; and Stokes' Papers, Vol. I., p. 230 "On the resistance of a 
fluid to two oscillating spheres"). You can thus tell the motion 
exactly in the neighbourhood of two spheres vibrating to and fro 
provided the amplitudes of their vibrations are small in comparison 
with the distance between them ; and you can find the value of 
A for two spheres of any given radii and any given distance 
between them. For such a thing as a tuning fork, you could 
not, of course, work it out theoretically ; but I think it would be 
an interesting subject for junior laboratory work, to find it by 
experiment. 

I suppose you are all now familiar with the zero of sound in 
the neighbourhood of a tuning fork ; but I have never seen it 
described correctly anywhere. We have no easy enough theo- 
retical means of determining the inclination of the line going to 
the position of the ear for silence to the line joining the prongs ; 
but we readily see that it is dependent upon the proportions 
of the body. In turning the tuning fork round its axis, you 
can get with great nicety the position for silence; and a sur- 
prisingly small turning of the tuning fork from the position of 
silence causes the motion to be heard. It would be very curious 
to find whether the position of zero sound varies perceptibly with 
the amplitude of the vibrations. I doubt whether any perceptible 
difference will be found in any ordinary case however we vary the 
amplitude of the vibrations. But I am quite sure you will find 
considerable difference, according as you take tuning forks with 
cylindrical prongs, or with rectangular prongs of such proportions 
as old Marlowe used to make, or tuning forks like the more 
modern ones that Koenig makes, with very broad flat prongs. 

Now for our molecular problem. Molecular 

I want to see how the variable quantities vary, when we vary 
the period. Remember that 



54 



LECTURE V. 



da. /.-)\ 

Molecular, go that = W ' ........................ 

Write for the 
for u { ; we find 



Write for the moment 8 for rf , _, , and differentiate the equation 



Substitute successively, and we find, 



This is our expression, and remark the exceedingly important 
property of it that it is essentially positive, i.e. the variation of . 
with increase of T~* is essentially positive. Now 



also -*i = n* and so on. 

M r x i u * X M 

The result (3) therefore is equivalent to the following expression 
(4) for the differential coefficient of u 4 with respect to the period. 



This is certainly a very remarkable theorem, and one of great 
importance with reference to the interpretation of the solution of 
our problem. Remember that x t is the displacement of m ( at any 
time of the motion. You may habitually think of the maximum 
values of the displacements, but it is not necessary to confine your- 
selves to the maximum values. Instead of x lt x t , ... x, we may 
take constants equal to the maximum values of the xs, each 

multiplied into sin , because the particles vibrate according to 

the simple harmonic law, all in the same period and the same 
phase, that is all passing through zero simultaneously, and reach- 
ing inaximums simultaneously, every vibration. The masses are 
positive, and we have squares of the displacements in the several 
terms of (4) ; so that the second member of (4) is essentially 



VIBRATIONS OF SERIAL MOLECULE. 55 

negative. Hence, as we augment the period, each one of the ratios Molecular 
u f decreases. 

Let us now consider the configurations of motion in our spring 
arrangement, for different given periods of the exciting vibrator, P. 
I am going to suppose, in the first place, that the period of 
vibration is very small, and is then gradually increased. As you 
increase the period, we have seen that the value of each one of the 
quantities u 1} u 2 , ... decreases. It is interesting to remark that 
this is so continuously throughout each variety of the configura- 
tions found successively by increasing r from to oo . But we 
shall find that there are critical values of r, at which one or other 
of the us, having become negative decreases to oo ; then suddenly 
jumps to + oo as r is augmented through a critical value; and 
again decreases, possibly again coming to + oo , possibly not, while 
r is augmented farther and farther, to infinity. In the first place, 
r may be taken so small that the us are all very large positive 

quantities; for u. being equal to ^c. c i+l , may be cer- 
tainly made as very large positive as we please by taking r small 
enough, if, at the same time the succeeding quantity, w. +1 , is large, 
a condition which we see is essentially fulfilled where r is very 
small, because we have u } = m,./T 2 Cj c j+l , which makes iij very 
great. 

Observe that the us all positive implies that , x v , x z , x 3 , ... x f 
are alternately positive and negative. In other words the handle 
P and the successive particles m,, m >2 , ... m j} are each moving in a 
direction opposite to its neighbour on either side. Since the 
magnitudes of the ratios u v u, 2 , ...Uj of the successive amplitudes 
decrease with the increase of the period, the amplitude of particle 
m. is becoming smaller in proportion to the amplitude of the suc- 
ceeding particle ra j+1 , as long as the vibrations of the successive 
particles are mutually contrary- wards. I am going to show you 
that as every one of these quantities u. decreases, the first that 
passes through zero is necessarily u^ ; corresponding to a motion of 
each particle of the system infinitely great in comparison with 
the motion of the handle P ; that is to say, finite simple harmonic 
motion of the system with P held fixed. This is our first critical 
case. It is the only one of the j fundamental modes of vibration of 
the system with P fixed, in which the directions of vibration of the 
successive particles are all mutually contrariwise, and it is the one 



56 LECTURE V. 

Molecular, of them of which the period is shortest. After that, us we in- 
crease T, ^ becomes negative, and the motion of P comes to be in 
the direction of the motion of the first particle. As we go on 
increasing the period we shall find that the next critical case that 
comes is one in which particle ?/i, has zero motion, or 



To prove this, and to investigate the further progress, let us look 
at the state of things when a positive decreasing w. has approached 

* 

very near to zero. We shall have U M , being equal to a^, - , a 

very large negative quantity. This alone shows that M^, must 
have preceded u ( in becoming zero, since it must have passed 
through zero before becoming negative. Therefore, as we augment 

T, the first of the 's to become zero is u t = -&- ; or, as I said 

37 

before, the motion of particle ?/&, and also of each of the other 
particles is infinite in comparison with the motion of P. Just 
before this state of things all the particles P, w,, ... w, are, as we 
saw, moving each contrary-wards to its neighbour; just after it, P 
has reversed its motion with reference to the first particle, and is 
moving in the same direction with it. 

This continues to be the configuration, till just before the 
second critical case, in which we have , large negative, w s small 
positive, 3 , ... Uj, all positive. At this critical case, we have 

u t = 2&. = + x ; or #, = x 

~"~ *^1 

The period of motion of P that will produce this state of things is 
equal to the period of the free vibration of the system of particles, 
with mass w t held at rest, and each of the other masses moving 
contrary-wards to its neighbour on each side. When the period of 
the simple harmonic motion of P is equal to a period of motion 
of the system with the first particle held at rest, then the only 
simple harmonic motion which the system with all the particles 
unconstrained can have is in that period, and with the amplitude 
of vibration of the second particle in one direction just so great as 
to produce by spring No. 2, a pull in that direction, on ?/i,, equal 
to the pull exercised on it through spring No. 1, by P in the 
opposite direction ; so as to let the first particle be at rest. Sup- 



VIBRATIONS OF SERIAL MOLECULE. 57 

pose now T to be continuously increased through this critical value. Moleculai 
Immediately after the critical case, u l has changed from large 
negative, through + ab , to large positive, and w 2 from small positive, 
through f 0, to small negative ; or the first particle has reversed 
the direction of its motion and come to move same-wards with P 
and contrary-wards to the second particle. 

The third critical case might be that of the second particle 
coming to rest, (u. 2 = + x> , u a = 0); or it might be w t = a second 
time : it must be either one or other of these two cases. But we 
must not stop longer on the line of critical cases at present*. I 
will just jump over the remaining critical cases to the final con- 
dition. 

It would be curious to find the solution when the period is 

infinitely great out of our equations. When T is infinite, ^ 

[Note added; Jan. 11, 1886, Netherhall, Largs.] 

* As we go on increasing T from the first critical value (that which made M a = 0), 
the essential decreasings of i^ (negative) and u, 2 (still positive) bring u 1 to - oo and 
2 to simultaneously. With farther increase of T, the decreasings of it. 2 (now 
negative) and of u 3 (still positive) bring H 2 to - oo and u 3 to 0, simultaneously; and 
so on, in succession from u t to u, which passes through zero to negative, but cannot 
become - oo and remains negative for all greater values of T, diminishing to the 
value - (Cj + c, +1 ) as r is augmented to infinity. 

But u v after decreasing to - x , must pass to + oo and again become decreasing 
positive. It must again pass through zero; and thus there is started another 
procession of zeros along the line from P, through m lt >.,,... successively, but ending 
in m,-_ 3 : not in ,-_! whose amplitude (M/O//C,) is made zero and negative by the 
conclusion of the first procession, before the second procession can possibly reach 
it. Thus H > _ I passes a second and last time through zero and diminishes to the 

limiting values - c,_-, ( -- 1 --- 1 -- )/( -- 1 -- ) , as r is augmented to oo . 

\ C J+l C J C J-lJ' \ C J-2 C J-l/ 

A third procession of zeros, similarly commencing with P, passing along the 
line, m lt 7/1.,,... and ending with m,_ 3 , makes w,_ 2 zero for a third and last time, and 
leaves it to diminish to its limit (shown by the formula reported in the text below, 
from the lecture,) as T augments to oo . Similarly procession after procession, in 
all j processions, commence with m r The last begins and ends in ;H I , and leaves MJ 
to go from zero to its negative limiting value 



as T augments to oo . There is no general rule of precedence in respect to magni- 
tude of T, of the different transitional zeros of the different processions. For 
example, there is no general rule as to order of commencement of one procession, 
and termination of its predecessor. The one essential limitation is that no 
collision can take place between the front of one procession and the rear of its 



58 LECTURE V. 

Molecular, vanishes, and a, c,- c. That applied to the equations for the 
it's ought to find the solution quite readily. 

You know, when you think of the dynamics of this case, that 
when r is infinitely great, P is moving infinitely slowly, so that 
the inertia of each particle has no sensible effect ; and all the 
particles are in equilibrium. Let F be the force, then, on the 
spring ; that is to say, pull P slowly down with a force F and hold 
it at rest. What will be the displacements of the different 

j~j J7t 77F 

particles ? Answer, a;, = , ^ = + ~ , and so on. Particle 

c j+i c J+i j 

number j is displaced to a distance equal to the force, divided by 
the coefficient of elongation of the spring. To obtain the displace- 
ment of particle j 1, we have to add the displacement resulting 
from the elongation of the next spring c ; , and so on. The general 
equation then is 



1 1 

4- 



c,,. 



(i, + -^ 



It is a curious but of course a very simple and easy problem to 
substitute the value of a. = - c i c i+l in the continued fraction 
which gives u., and verify this solution. No more of this now 
however. 

It is fiddling while Rome is burning ; to be playing with 
trivialities of a little dynamical problem, when phosphorescence 
is in view, and when explanation of the refraction of light in 
crystals is waiting. The difficulty is, not to explain phosphor- 
escence and fluorescence, but to explain why there is so little of 
sensible fluorescence and phosphorescence. This molecular theory 
brings everything of light, to fluorescence and phosphorescence. 
The state of things as regards our complex model-molecule would 
be this : Suppose we have this handle P moved backwards and 
forwards until everything is in a perfectly periodic state. Then 
suddenly stop moving P. The system will continue vibrating for 
ever with a complex vibration which will really partake something 
of all the modes. That I believe is fluorescence. 

But now comes Mr Michelson's question, and Mr Newcomb's 
question, and Lord Rayleigh's question ; the velocity of groups 
of waves of light in gross matter. Suppose a succession of luminous 



FINITE SEQUENCES OF WAVES IN A DISPERSIVE MEDIUM. 59 

vibrations commences. In the commencement of the luminous vi- Molecular, 
orations the attached molecules imbedded in the luminiferous ether, 
do not immediately get into the state of simple harmonic vibration 
which constitutes a regular light. It seems quite certain that 
there must be an initial fluorescence. Let light begin shining on 
uranium glass. For the thousandth of a second, perhaps, after 
the light has begun shining on it, you should find an initial state 
of things, which differs from the permanent state of things some- 
what as fluorescence differs from no light at all. 

There is still another question, which is of profound interest, 
and seems to present many difficulties, and that is, the actual 
condition of the light which is a succession of groups. Lord 
Rayleigh has told us in his printed paper in respect to the 
agitated question of the velocity of light, and then again, at the 
meeting of the British Association at Montreal, he repeated very 
peremptorily and clearly, that the velocity of a group of waves 
must not be confounded with the wave velocity of an infinite 
succession of waves, and is of necessity largely different from the 
velocity of an infinite succession of waves, in every dispersively 
refracting medium, that is to say, medium in which the velocity 
is different for lights of different period. It seems to be quite 
certain that what he said is true. But here is a difficulty which 
has only occurred to me since I began speaking to you on the 
subject; and I hope, before we separate, we shall see our way 
through it. All light consists in a succession of groups. We are 
all, already, familiar with the question ; Why is all light not 
polarized ? and we are all familiar with its answer. We are now 
going to work our way slowly on until we get expressions for 
sequences of vibrations of existing light. Take any conceivable 
supposition as to the origin of light, in a flame, or a wire made 
incandescent by an electric current, or any other source of light; 
we shall work our way up from these equations which we have 
used for sound, to the corresponding expression for light from 
any conceivable source. Now, if we conceive a source con- 
sisting of a motion kept going on with perfectly uniform period- 
icity ; the light from that source would be plane polarized, or 
circularly polarized, or elliptically polarized, and would be abso- 
lutely constant. In reality, there is a multiplicity of successions 
of groups of waves, and no constant periodicity. One molecule, 
of enormous mass in comparison with the luminiferous ether that 



60 LECTURE V. 

Molecular, it displaces, gets a shock and it performs a set of vibrations until 
it comes to rest or gets a shock in some other direction ; and it 
is sending forth vibrations with the same want of regularity that 
is exhibited in a group of sounding bodies consisting of bells, 
tuning forks, organ pipes, or all the instruments of an orchestra 
played independently in wildest confusion, every one of which is 
sending forth its sound which, at large enough distances from the 
source, is propagated as if there were no others. We thus sec 
that light is essentially composed of groups of waves ; and if the 
velocity of the front or rear of a group of waves, or of the centre 
of gravity of a group, differs from the wave- velocity of absolutely 
continuous sequences of waves, in water, or glass, or other dispcrsivcly 
refracting mediums, we have some of the ground cut from under us 
in respect to the velocity of waves of light in all such mediums. 

I mean to say, that all light consists of groups following one 
another, irregularly, and that there is a difficulty to see what to 
make of the beginning and end of the vibrations of a group : and 
that then there is the question which was talked over a little in 
Section A at Montreal, will the mean of the effects of the 
groups be the same as that of an infinite sequence of uniform 
waves, and will the deviation from regular periodicity at the 
beginning and end of each group have but a small influence on 
the whole ? It seems almost certain that it must have but a 
small influence from the known facts regarding the velocity of 
light proved by the known, well-observed, and accurately measured, 
phenomena of refraction and interference. But I am leading 
you into a muddle, not however for you, I hope, a slough of 
despond; though I lead you into it and do not show you the 
way out. You will all think a good deal along with me about the 
connections of this subject. 



LECTURE VI. 
MONDAY, Oct. 6, 5 P.M. 

I WANT to ask you to note that when I spoke of k + $n not 
differing very much from k for most solids, I was rather under the 
impression for the moment that the ratio of n to k was smaller 
than it is; and also you will remember that we had + J^j on 
the board by mistake for k + ;. The square of the velocity of a 
condensational wave in an elastic solid is(Jk + fn)/p. For solids 
fulfilling the supj>osed relation of Navier and Poisson between 
compressibility and rigidity we have n = $&; and for such cases the 
numerator becomes %k. It would be k if there wore no rigidity ; 
it is k if the rigidity is that of a solid for which Poisson ' ratio 
has its supj)osed value. 

Metals are not enormously far from fulfilling this condition, 
but it seems that for elastic solids generally n bears a less pro- 
portion to /. than this. It is by no means certain that it fulfils it 
even approximately for metals ; and for india rubber, on the other 
hand, and for jellies, n is an exceedingly small fraction of k, so 
that in these cases the velocity of the condensational wave is but 

/k 
very little in excess of /-. The velocity of propagation of a 



distortional wave is *-; so that for jellies, the velocity of propa- 

gation of condensational wave is enormously greater than that of 
distortional waves. 

I am asked by one of you to define velocity potential. Those 
who have read German writers on Hydrodynamics already know 
the meaning of it perfectly well. It is a purely technical expres- 
sion which has nothing to do with potential or force. " Velocity 
potential " is a function of the co-ordinates such that its rate of 
variation per unit distance in any direction is equal to the 



62 LECTURE VI. PART 1. 

Molar. component of velocity in that direction. A velocity potential 
exists when the distribution of velocity is expressible in this way ; 
in other words when the motion is irrotational. The most con- 
venient analytical definition of irrotational motion is, motion such 
that the velocity components are expressed by the differential co- 
efficients of a function. That function is the velocity potential. 
When the motion is rotational there is no velocity potential. 

This is the strict application of the words " velocity potential " 
which I have used. A corresponding language may be used for 
displacement potential. It is not good language, but it is con- 
venient, it is rough and ready. And when we are speaking of 
component displacements in any case, whether of static displace- 
ment in an elastic solid or of vibrations, in which the components 
of displacement are expressible as the differential coefficients of a 
function, we may say that it is an irrotational displacement. If from 
the differentiation of a function we obtain components of velocity, 
we have velocity potential ; whereas, if we so get components of 
displacement, we have displacement potential. The functions <j>, 
that we used, are not then, strictly speaking, velocity potentials 
but displacement potentials. 

I want you in the first place to remarK what is perfectly well 
known to all who arc familiar with Differential equations, that 

taking the solution = - . sin q as a primary, where 





we may derive other solutions by differentiations with respect to 
the rectangular co-ordinates. The first thing I am going to call 
attention to is that at a distance from the origin, whatever be 
the solution derived from this primary by differentiation, the cor- 
responding displacement is nearly in the direction through the 
origin of co-ordinates. 

Take any differential coefficient whatever -f' +>+ * A; the term 

ct oc d \i ciz 

of this which alone is sensible at an infinitely great distance is 
that which is obtained by successive differentiation of sin q. That 
distance term in every case is as follows : 



dy '\dz 



CONDENSATIONAL WAVE. 63 

It will be sin q or cos</, according as i+j + k is even or odd. We Molar, 
do not need to trouble ourselves about the algebraic sign, because 
we shall make it positive, whether the differential coefficient is 

XT d r x dr y dr z ml 

positive or negative. Now =-, -,-=, -=-=-. Thus our 
ax r ay r dz r 



type solution l>ecomes, 4^^-n ( l- This expresses the most 

general type of displacement potential for a condensational wave 
proceeding from a centre, and having reached to a distance in 
any direction from the centre, great in comparison with the wave- 
length. I have not formally proved that this is the most general 
type, but it is very easy to do so. I am rather going into the 
thing synthetically. It is so thoroughly treated analytically by 
many writers that it would be a waste of your time to go into 
anything more, at present, than a sketch of the manner of treat- 
ment, and to give some illustrations. 

But now to prove that the displacement at a distance from the 
origin of the disturbance is always in the direction of the radius 
vector. Once more, the differential coefficient of this displace- 
ment potential, which has several terms depending upon the 
differentiation of the r's, ar's, etc. has one term of paramount 

importance, and that Is the one in which you get as a factor. 

A, 

The smallness of X in proportion to the other quantities makes 
the factor give importance to the term in which it Is found. 

\ 

The distance terms then for the components of the displacement 
are 

fc _*VY 2,r scos *cos 



r sin 



These are then the components of a displacement which is radial ; 
and the expression for the amplitude of the radial displacement is 



The sum of any number of such expressions will express the 
distance effect of sound proceeding from a source. It is interest- 



64 LECTURE VI. PART I. 

ing to see how, simply by making up an algebraic function in the 
numerator out of the x\ ys and z's, we can get a formula that 
will express any amount of nodal subdivision where silence is felt. 
The most general result for the amplitude of the radial displace- 

ment is E = 2 !- Remark that * , , Z - are merely angular 






functions and may be expressed at once as sin cosi/r, sin sin i/r, 
cos0; and therefore R is an integral algebraic function of sin0cos^, 
sin sin ^, cos 0. It is thus easy to see that you can vary inde- 
finitely the expressions for sound proceeding from a source with 
cones of silence and corresponding nodes or lines in which those 
cones cut the spherical wave surface. It is interesting to see that 
even in the neighbourhood of the nodes the vibration is still 
perpendicular to the wave surface; so that we have realized in 
any case a gradual falling off of the intensity of the wave to zero 
and a passing through zero, which would be equivalent to a change 
of phase, without any motion perpendicular to the radius vector. 

The more complicated terms that I have passed over are those 
that are only sensible in the neighbourhood of the source. Sup- 
pose, for instance, that you have a bell vibrating. The air slipping 
out and in over the sides of the bell and round the opening gives 
rise to a very complicated state of motion close to the bell ; and 
similarly with respect to a tuning fork. If you take a spherical 
body, you can very easily express the motion in terms of spherical 
harmonics. You see that in the neighbourhood of the sounding 
body there will be a great deal of vibration in directions perpen- 
dicular to the radius vector, compounded with motions out and in ; 
but it is interesting to notice that all except the radial component 
motions become insensible at distances from the centre large in com- 
parison with the wave-length. It is the consideration of the motion 
at distances that are moderate in comparison with the wave-length 
that Stokes has made the basis of that very interesting investiga- 
tion with reference to Leslie's experiment of a bell vibrating in a 
vacuum, to which I have already referred. (Lecture in., pp. 30, 37 
above.) 

We may just notice, before I pass away from the subject, two 
or three points of the case, with reference to a tuning fork, a bell, 
and so on. Suppose the sounding body to be a circular bell. In 
that case clearly, if the bell be held with its lip horizontal, and if 
it be kept vibrating steadily in its gravest ordinary mode, the 



VIBRATIONS OF A BELL. 65 

kind of vibration will be this : a vibration from a circular figure Molar. 

U^-H into an elliptic figure (Lx^J) along one diameter, and 
\JL-X 

a swinging back through the circular figure \*"T y > n t an 
elliptic figure (f X )) a ' on g the diameter at right angles to the 




first Clearly there would be practically a plane of silence 
here and another at right angles to it here (represented on the 
diagrams by dotted lines). Hence the solution for the radial 
component corresponding to this case, at a considerable distance 

from the bell, is R = ( \ cosM) ^ , in order that the component 

may vanish when cosM = 4, or A = 45 ; A being an azimuthal 
angle if the axis of the bell is vertical. 

On the other hand, consider a tuning-fork vibrating to-and-fro 
or an elongated (elliptic) bell, which I got from that fine old 
Frenchman, Koenig's predecessor, Marloye. It makes an ex- 
ceedingly loud sound and has an advantage in acoustic experi- 
ments over a circular bell. If you set a circular bell vibrating 
and leave it to itself you always hear a beating sound, because 
the bell is approximately but not accurately symmetrical. Excite 
it with a bow, and take your finger off, and leave it to itself: and 
if you do not choose a proper place to touch it, for a fundamental 
mode, when you take your finger off it will execute the re- 
sultant of two fundamental modes. 

I do not know whether the corresponding experiment with 
circular plates is familiar to any of you. I would be glad to know 
whether it is. I make it always before my own classes, in illus- 
trating the subject. Take a circular plate just one of the ordinary 
circular plates that are prepared for showing vibration in acoustic 
illustrations. Excite it in the usual way with a violoncello bow, and 
putting a finger, or two fingers, to the edge to make the quadrantal 
vibration. If sand is sprinkled on the plate, the vibrations toss it 
into sand-hills with ridges lying along two diameters of the disc, 
perpendicular to one another, one of them through the point or 
T. L. 5 



(56 LECTURE VI. PART 1. 

Molar. two of them through the points of the edge touched by finger. Now 
cease bowing and take your finger off the edge of the disc. The 
sand-hills are tossed up in the air and the sand is scattered to 
considerable distances on each side, and is continually tossed up 
and not allowed to rest anywhere. At the same time you hear a 
beating sound. But by a little trial, I find one place where if I 
touch with the finger, and ply the bow so as to make a quadrantal 
vibration, and if I then cease bowing and take off my finger, the 
sand remains undisturbed on two diameters at right angles, and no 
beat is heard. Then having found one pair of nodal diameters, 
I know there will be another pair got by touching the plate here, 
45 from the first place. Now touch therefore with two fingers, at 
two points 90 from one another midway between the first and 
second pairs of nodal diameters : cause the plate to vibrate and 
then take off your fingers, and stop bowing : you will hear very 
marked beats ; a sound gradually waxing from absolute silence 
to loudest sound, then gradually waning to silence again, and so 
on, alternating between loudest sound and absolute silence with 
perfect gradualness and regularity of waxing and waning. 

Take a division of the circumference into six equal parts by 
three diameters, and you find the same thing over again. Go on 
by trial touching the plate at two points GO or 120 asunder, and 
bowing it 30 from either; and you will see the sand resting 
on the three diameters determined by your fingers. Take off 
your fingers and you will in general see the sand scattered and 
hear a beat. Follow your way around, little by little it is very 
pretty when you come near a place of no beat. The moment you 
take off your fingers you see the lines of nodes swaying to-and- 
fro on each side of a mean position, with a slow oscillation; 
and you hear a very distinct beat, though of a soft but perfectly 
regular character from loudest to least loud sound. Get exactly 
the mean position, steadying the nodal sand-hills while still plying 
the bow : then cease bowing and suddenly remove your finger or 
fingers from the plate ; you will see the nodal lines remaining 
absolutely still and you will hear a pure note without beats. If 
you touch at exactly 30 from the nodal lines first found you will 
have the strongest beat possible, which is a beat from loud sound 
to silence. Advance your fingers another 30 and you will again find 
the sand-hills remain absolutely still when you remove your fingers. 
You may go on in this way with eight and ten subdivisions, and so 



VIBRATIONS OF APPROXIMATELY CIRCULAR PLATES. 67 

on ; but you must not expect that the places for the sextan tal, Molar, 
octantal, and higher, subdivisions correspond to the places for the 
quadrantal subdivisions. The places for quadrnntal subdivision 
will not in general be places for octantal subdivision. You must 
experiment separately for the octantal places, and you will find 
generally that their diameters are oblique to the quadrantal. 

The reason for all this is quite obvious. In each case, the 
plate being only approximately circular and symmetrical, the 
general equation for the motion has two approximately equal 
roots corresponding to the nodes or divisions by one, two, three, or 
four diameters, and so on. Those two roots always correspond to 
sounds differing a little from one another. The effect of putting 
the finger down at random is to cause the plate, as long n.s your 
finger is on it, to vibrate forcedly in a simple harmonic vibration 
of period greater than the one root and less than the other. But 
as soon as you take your finger off, the motion of the plate follows 
the law of superposition of fundamental modes ; each fundamental 
mode being a simple harmonic vibration. I have often, in showing 
this experiment, tried musicians with two notes which were very 
nearly equal, and said to them, " Now, which of the two notes is 
the graver?" Rarely can they tell. The difference is generally too 
small for a merely musical ear, and the verdict is that the notes 
are "the same ;" musicians are not accustomed to listen to sounds 
with scientific ears and do not always say rightly which is the 
graver note, even when the difference is perceptible. Any person 
can tell, after having made a few experiments of the kind, that 
this is the graver and that the less grave note, even though he 
may have what, for musicians, is an uncultivated ear, or truly a 
very bad ear for music, not good enough in fact to guide him in 
sounding a note with his voice, or to make him sing in tune if he 
tries to sing. It is very curious, when you have two notes which 
you thoroughly know are different, that if you sound first one and 
then the other, most people will say they are about the same. 
But sound them both together, and then you hear the discord of 
the two notes in approximate unison. 

In every case of a circular plate vibrating between diametral 
lines of nodes, there is an even number of planes of silence in the 
surrounding air; being the planes perpendicular to the plate, 
through the nodal diameters. If you take a square plate or bell 
vibrating in a quadrantal mode, for instance, then you have two 

52 



68 LECTURE VI. PART I. 

vertical planes of silence at right angles to one another. If you 
make it vibrate with six or more subdivisions, you will have a 
corresponding number of planes of silence. 

With reference to the motions in the neighbourhood of the 
tuning-fork, you get this beautiful idea, that we have essentially 
harmonic functions to express them. Essentially algebraic func- 
tions of the co-ordinates appear in these distant terms, but in the 
other terms which Prof. Stokes has worked out, and which have 
been worked out in Prof. Kowland's paper on Electro-Magnetic 
Disturbances*, quite that kind of analysis appears, and it is most 
important. I have not given you a detailed examination of that 
part of our general solution, but only called your attention specially 
to the "distance terms,' partly because of their interest for sound 
and partly because the consideration of them prepares us for our 
special subject, waves of light. 

To-morrow we shall begin and try to think of sources of waves 
of light. I want to lead you up to the idea of what the simplest 
element of light is. It must be polarized, and it must consist of 
a single sequence of vibrations. A body gets a shock so as to 
vibrate ; that body of itself then constitutes the very simplest 
source of light that we can have ; it produces an element of light. 
An element of light consists essentially in a sequence of vibrations. 
It is very easy to show that the velocity of propagation of 
sequences in the pure luminiferous ether is constant. The sequence 
goes on, only varying with the variation of the source. As the 
source gradually subsides in giving out its energy the amplitude 
evidently decreases ; but there will be no throwing off of wavelets 
forward, no lagging in the rear, no ambiguity as to the velocity of 
propagation. But when light, consisting as it does of sequences 
of vibrations, is propagated through air or water, or glass, or 
crystal, what is the result ? According to the discussion to which 
I have referred, the velocity should be quite uncertain, depending 
upon the number of waves in the sequence, and all this seems 
to present a complicated problem. 

But I am anticipating a little. We shall speak of this here- 
after. One of you has asked me if I was going to get rid of the 
subject of groups of waves. I do not see how we can ever get rid 
of it in the wave theory of light. We must try to make the best 
of it, however. 

* Phil. Mag. xvn., 1881, p. 413. Am. Jour. Math. \i., 1881, p. 350. 



NUMBER OF TERMS IN DETERMINANT. G9 

Ten minutes' interval. 

This question of the vibration of connected particles is a Molecular. 
peculiarly interesting and important problem. I hope you are 
not tired of it yet. You see that it is going to have many ap- 
plications. In the first place remark that it might be made the 
base of the theory of the propagation of waves. When we take 
our particles uniformly distributed and connected by constant 
springs we may pass from the solution of the problems for the 
mutual influence of a group of particles to the theory, say, of the 
longitudinal vibrations of an elastic rod, or, by the same analysis, 
to the theory of the transverse vibrations of a cord. 

I am going to refer you to Lagrange's Mtcanique Analytfque, 
[Part II. p. 339]. The problem that I put before you here is given 
in that work under the title of vibrations of a linear system of 
bodies. Lagrange applies what he calls the algorithm of finite 
differences to the solution. The problem which I put before you 
is of a much more comprehensive kind ; but it is of some little 
interest to know that cases of it may be found, ramifying into each 
other. 

I wish to put before you some properties of the solution which 
are of very great importance. I want you to note first the number 
of terms. 

We have : 



a/i, 
&. 



All the it-'s being Expressed in this way successively in terms of x r 
Let iV \ be the number of terms in x t , c These terms are obtained 
by substituting the values of x t . M , :r,_ H , in the formula 

~~ 9/-H-1 X i . i = Bf-tti^r-H-i ' ^/-m^v-t+r 

None of the terms can destroy one another except for special 
values, and the conclusion is that we have the following formula 
for obtaining the number of terms : 



This is an equation of finite differences. Apply the algorithm 



70 LECTURE VI. PART I. 

Molecular, of finite differences, as Lagrange says ; or, which is essentially the 
same, we may try for solutions of this equation by the following 
formula : N t = zN t . v We thus find 

l+v/5 
z* = z + 1, or z = 9 

We can satisfy our equation by taking either the upper or the 
lower sign. The general solution is, of course, 

/I +N/5Y /l-N/oV 



where C, G' are to be determined by the equation N =l, N t =1. 
It is rather curious to see an expression of this kind for the 
number of terms in a determinant. You will find that, of the more 
general equation 



the following is a solution: 



where r, s are the two roots of the equation ./" a.f 4- b. Remark 
that the coefficients of N , N it being symmetrical functions of the 
two roots, are, as they must of course be, integral functions of a 
and b. 

If one of the roots, s, for example, be less than unity, we may 
omit the large powers of s, and therefore for large values of i we 
may be sure of obtaining N, to within a unit, and therefore the 
absolutely correct value, by calculating the integral part of 



I 

r+ - 
r 

It is interesting to remark that the numerical value of this 
formula differs less and less from an integer the greater is i, and 
differs infinitely little from an infinitely large integer when i is 
infinitely great, 

The values of N t up to /=12 for the case of our problem 
(a, = l)=N = N l = \') are, 

t = 2, 8, 4, :>, (i, 7, S, 0, 10, 11, 12. 
N t = :>, 8, 5, S, l:{, 21, 84, 55, 81), 144, 233. 



LECTURE VII. 
TUESDAY, Oct. 7, 3.30 I-.M. 

LACKANOE, in the second section of the second part of his Me- Moleculi 
canique Analytique on the Oscillation of a Linear System of Bodies, 
has worked out very fully the motion in the first place for bodies 
connected in series, and secondly for a continuous cord. The case 
that we are working upon is not restricted to equal masses and 
equal connecting springs, but includes the particular linear system 
of Lagrange, in which the masses and springs are equal. I hope to 
take up that particular case, as it is of great interest. We shall 
take up this subject first to-day, and the propagation of disturbances 
in an elastic solid second. 

It was pointed out by l)r Franklin that, for the particular case 
^V, = aN v which is the case of our particular question as to the 
number of terms in our determinant, the formula becomes 



/ S-J . r'-'-s'-'N xr 
(a - +6 - LV. 
V r-s r-s J 



and may be thus simplified. 

We have r" = ur + 6, or multiplying by r' ', r** 1 = ar 1 + br*' 1 . So 
that the expression simplifies down to 



This may be obtained directly, by determining C, (7, in terms of 
A., with M, = 0, to make N t = Cr* + CV. 

We have, in our case, a = 6 = 1 : whence 

r - s = s/5, r = l -~ = 1-618, = - "618. 



72 LECTURE VII. PART 1. 

Molecular. If we work this out by very moderate logarithms for the case 

*.--.. 

dropping s 13 , we find 

13 log 1-618 - logv/5 = 13 x -209 - '3495 = 2'3675 = log 233, 

which comes out exact ; and this working with only 4-place- 
logarithms. 

I want to call your attention to something far more important 
than this. The dynamical problem, quite of itself, is very interest- 
ing and important, connected as it is with the whole theory of 
modes and sequences of vibration ; but the application to the 
theory of light, for which we have taken this subject up, gives to 
it more interest than it could have as a mere dynamical problem. 
I want to justify a fundamental form into which we can put our 
solution, which is of importance in connection with the application . 
we wish to make. 

Algebra shows that we must be able to throw ' 1 into the form 




where ry,, </,...</, are determinate constants, and K V ,...*, are the 
values of the period T for which -- -* becomes infinite. We can 

put it into this form certainly, for if .r, and be expressed in terms of 
<KJ, they will be functions of the (j - If and /' degrees, respectively, 

in -5. This is easily seen if we notice that av_, = - ^ x. is of the 
first degree in -5, and that the degree of each x is raised a unit 
above that of the succeeding x by the factor 0, = - c, c H1 in 

the equation - cp^ = < w + c i+l <x i+1 . Therefore, writing z for \, 
we have 



SOLUTION IN TERMS OF ROOTS. 73 

This, on being expanded into partial fractions, becomes Molecul* 

K.. A.. K.. 




which takes the previously written form if we put K^cf q f 

We know that the roots of the equation of the I th degree in z 
which makes ' become infinite are all real ; they are the 

periods of vibration of a stable system of connected bodies. We 
have formal proof of it in the work which we have gone through 
in connection with such a system. I am putting our solution in 
this form, because it is convenient to look upon the characteristic 
feature of the ratio of T to one or other of the fundamental 
periods. In the first place it is obvious that if we know the roots 
* t , ic f ,..., the determination of </,, 9,,... is algebraic. Another form 
which I shall give you is an answer to that algebraic question, 
what are the values of q lt y t ... ? It is an answer in a form that is 
jwirticularly appropriate for our consideration because it introduces 
the energy of the vibrations of the several fundamental modes in a 
remarkable manner. We will just get that form down distinctly. 

Take the differential coefficients of ' with respect to 

-, , and denoting J 1 , \ - 1 , for brevity, by D v D r we 

find 

d c. 



For the case T = ,, our differential coefficient becomes - l , which 
determines 

Now you will remember that we had 



For the moment, take the expression for the simple harmonic 
motion, and you see at once that that conies out in terms of the 
energy. Adopt the temporary notation of representing the 



74 LECTUKE VII. PART I. 

Molecular, maximum value by an accented letter. Then we have at any time 
of the motion # t = x\ sin , if we reckon our time from an era of 

each particle passing through its middle position, remembering 
that all the particles pass the middle position at the same instant. 
We have therefore for the velocity of particle No. 1, 



= X , COS 

T T 



The energy which at any time is partly kinetic and partly 
potential, will be all kinetic at the instant of passing through the 
middle position. Take then the energy at that instant. For t = 

we have x 1 = 0, J\ = x' v Denoting the whole energy by E 
(and remembering that the mass = ',J we have 



Thus, the ratio of the whole energy to the energy of the first 
particle / - '-r-t \ being denoted by 2i~ l , we have 



>article / - r, \ beiny deno 

i '";:) 



uv ^ ss j^o i 

(IT ,7'j 

This is true for any value of T whatever. From this equation find 
then the ratios of the whole energy to the energy of the first 
particle when T = K I , .,,.... Denoting these several ratios by H~ l , 

R' 1 ..., we find J/^- 1 l - , </ 2 = 2 , .... Our solution becomes 
then 

~3~tfi*+&+- 

lS> i \-| K^ T 

This is a very convenient form, as it shows us everything in terms 
of quantities whose determinations are suitable, and intrinsically 
important and interesting, viz. the periods and the energy-ratios. 

It remains, lastly, to show how, from our process without cal- 
culating the determinants, we can get everything that is here 
concerned. Our process of calculating gives us the it's in order, 
beginning with WjLl . That gives us the a'a in order, and thus 
we have all that is embraced in the differential coefficient with 



DETERMINATION OF ROOTS. 75 

respect to -, . Everything is done, if we can find the roots. I Muloculi 

will show how you can find the roots from the continued fraction, 
without working out the determinant at all. The calculation in 
the neighbourhood of a root gives us the train of x's corresponding 
to that root, and then by multiplying the squares of the ratios of 
the xa to x v by the masses, and adding, we have the corresponding 
energy. 

The case that will interest us most will be the successive 
masses greater and greater; and the successive springs stronger 
and stronger, but not in proportion to the masses so that the 
periods of vibration of limited portions of the higher numbered 
particles of the linear system shall be very large : so that if we 
hold at rest particles 4 and G, the natural time of vibration of 
particle 5 will be longer than No. 2's would be if we held Nos. 1 
and 3 at rest and set No. 2 to vibrate, and so on. 

We will just put down once more two or three of our equations : 

c, c, 1 c * MI. 

= a, - *- ; . . . ; ti. = a. : a = ' c. c . , . 

-*. . "m T 

Without considering whether u^ is absolutely large or small, let 
us suppose that it is large in comparison with c t+I ; 
u 4 will then be of the order a t ; u f _ l of the order o^,; and so on. 
We are to suppose that a,, t , ...a, are in ascending order of 

magnitude. Now, M, t ... i< ( = ( )'c, ... c t . We thus have this 

important proposition, that the magnitudes of the vibrations of 
the successive particles decrease from particle No. 1 towards No. j; 
and x i is exceedingly small in comparison with , even though 
there is only a moderate proportion of smallness with respect to 

the ratios ^ , - 1 ,,... Thus see how small is the motion at a 

, "t u t 

considerable distance from the point at which the excitation is 
applied, under the suppositions that we have been making. 

Now, as to the calculations. I do not suppose anybody is 
going to make these calculations*, but I always feel in respect to 
arithmetic somewhat as Green has expressed in reference to 

Happily Urn negative prognostication was not fulfilled. See (Appendix C) 
Numerical Solution with Illustrative Curve*, by Prof. E. W. Morley, of the case of 
Bercn connected massed, proposed in Lecture IX 



76 LECTURE VII. PART I. 

Molecular, analysis. I have no satisfaction in formulas unless I feel their 
arithmetical magnitude at all events when formulas are intended 
for definite dynamical or physical problems. So that if I do not 
exactly calculate the formulas, I would like to know how I could 
calculate them and express the order of the magnitudes concerned 
in them. We are not going to make the calculations, but you will 
remark that we have every facility for doing so. In the first 
place, is the exceeding rapidity of convergence of the formulas. 

The question is to find ^- ; everything, you will find, depends 

-x l 

upon that. The exceeding rapidity of the convergence is mani- 
fest. Since u 2 is large, u t is equal to a t with a small correction ; 
similarly u 2 = a 2 with a small correction, and so on ; so that two or 
three terms of the continued fraction will be sufficient for cal- 
culating the ratio denoted by u 1 . The continued fractions con- 
verge with enormous rapidity upon the suppositions we have been 
making. We thus know the value of the differential coefficient 

jr-<z\ We can in this way obtain several values of u t and begin 

to find it coming near to zero. Then take the usual process. 
Knowing the value of the differential coefficient allows you to 
diminish very much the number of trials that you must make for 
calculating a root. The process of finding the roots of this con- 
tinued fraction will be quite analogous to Newton's process for 
finding the roots of an algebraic equation ; and I tell any of you 
who may intend to work at it, that if you choose any particular 
case you will find that you will get at the roots very quickly. 

I should think something like an arithmetical laboratory would 
be good in connection with class work, in which students might be 
set at work upon problems of this kind, both for results, and in 
order to obtain facility in calculation. 

I hinted to you in the beginning about the kind of view that 
1 wanted to take of molecules connected with the luminiferous 
ether, and affecting by their inertia its motions. I find since then 
that Lord Rayleigh really gave in a very distinct way the first 
indication of the explanation of anomalous dispersion. I will just 
read a little of his paper on the Reflection and Refraction of Light 
by intensely Opaque Matter (Phil. Mag., May, 1872). He com- 
mences, " It is, I believe, the common opinion, that a satisfactory 
mechanical theory of the reflection of light from metallic surfaces 



RAYLEIGH ON CAUCHY'S METALLIC REFLECTION. 77 

has been given by Cauchy, and that his formula} agree very well MolecuUi 
with observation. The result, however, of a recent examination of 
the subject has been to convince me that, at least in the case of 
vibrations performed in the plane of incidence, his theory is 
erroneous, and that the correspondence with fact claimed for it is 
illusory, and rests on the assumption of inadmissible values for the 
arbitrary constants. Cauchy, after his manner, never published 
any investigation of his formulae, but contented himself with a 
statement of the results and of the principles from which he 
started. The intermediate steps, however, have been given very 
concisely and with a command of analysis by Eisenlohr (Pogg. 
Ann. Vol. Civ. p. 368), who has also endeavoured to determine 
the constants by a comparison with measurements made by Jamin. 
I propose in the present communication to examine the theory of 
reflection from thick metallic plates, and then to make some 
remarks on the action on light of a tliin metallic layer, a subject 
which has been treated experimentally by Quincke. 

" The peculiarity in the behaviour of metals towards light is 
supposed by Cauchy to lie in their op<icity, which has the effect of 
stopping a train of waves before they can proceed for more than a 
few lengths within the medium. There can be little doubt that 
in this Cauchy was perfectly right; for it has been found that 
bodies which, like many of the dyes, exercise a very intense 
selective absorption on light, reflect from their surfaces in ex- 
cessive proportion just those rays to which they are most opaque. 
Permanganate of potash is a beautiful example of this given by 
Prof. Stokes. He found (Phil. May., Vol. vi. p. 293) that when 
the light reflected from a crystal at the polarizing angle is ex- 
amined through a Nicol held so as to extinguish the rays polarized 
in the plane of incidence, the residual light is green ; and that, 
when analyzed by the prism, it shows bright bands just where 
the absorption-spectrum shows dark ones. This very instructive 
experiment can be repeated with ease by using sunlight, and 
instead of a crystal a piece of ground glass sprinkled with a little 
of the powdered salt, which is then well rubbed in and burnished 
with a glass stopper or otherwise. It can without difficulty be so 
arranged that the two spectra are seen from the same slit one 
over the other, and compared with accuracy. 

"With regard to the chromatic variations it would have seemed 
most natural to suppose that the opacity may vary in an arbitrary 



78 LECTURE VIE. PART I. 

Molecular, manner with the wave-length, while the optical density (on which 
alone in ordinary cases the refraction depends) remains constant, 
or is subject only to the same sort of variations as occur in trans- 
parent media. But the aspect of the question has been materially 
changed by the observations of Christiansen and Kimdt (Pogg. 
Ann. vols. CXLL, CXLIIL, CXLIV.) on anomalous dispersion in 
Fuchsin and other colouring-matters, which show that on either 
side of an absorption-band there is an abnormal change in the 
refrangibility (as determined by prismatic deviation) of such a 
kind that the refraction is increased below (that is, on the red 
side of) the band and diminished above it. An analogy may be 
traced here with the repulsion between two periods which frequently 
occurs in vibrating systems. The effect of a pendulum suspended 
from a body subject to horizontal vibration is to increase or diminish 
the virtual inertia of the mass according as the natural period of 
the pendulum is shorter or longer than that of its point of suspen- 
sion. This may be expressed by saying that if the point of support 
tends to vibrate more rapidly than the pendulum, it is made to go 
faster still, and vice versa " I cannot understand the meaning of the 
next sentence at all. There is a terrible difficulty with writers in 
abstruse subjects to make sentences that are intelligible. It is 
impossible to find out from the words what they mean ; it is only 
from knowing the thing* that you can do so "Below the absorp- 
tion-band the material vibration is naturally the higher, and hence 
the effect of the associated matter is to increase (abnormally) the 
virtual inertia of the aether, and therefore the refrangibility. On 
the other side the effect is the reverse." Then follows a note, "See 
Sellmeier, Pogg. Ann. vol. CXLIII. p. 272." Thus Lord Rayleigh 
goes back to Sellmeier, and I suppose he is the originator of all 
this. "It would be difficult to exaggerate the importance of these 
facts from the point of view of theoretical optics, but it lies 
beside the object of the present paper to go further into the ques- 
tion here." 

There is the first clear statement that I have seen. Prof. 
Rowland has been kind enough to get these papers of Lord 
Rayleigh for me. I am most grateful to him and others among 
you, by whom, with great trouble kindly taken for me, an 

In the next sentence, for "the refrangibility," substitute it* rrfructivity. 
W. T. Feb. 9, 1892. 



8ELLMEIER, HELMHOLTZ, LOMMEL. 79 

immense number of books have been brought to me, in every one Moleculai 
of which I have found something very important 

Sellmeier, Lord Rayleijjh, Helmholtz, and Lotmnel seem to be 
about the order. Lommel does not quote Helmholtz. I am rather 
surprised at this, because Lommel comes three or four years after 
Helmholtz: 1874 and 1878 are the respective dates. Lommel's 
paper is published in Helniholtz's Journal (Ann. der Physik und 
Chenrie, 1878, vol. III. p. 339), Helniholtz's paj>er is excellent. 
Lommel goes into the subject still further, and has worked uut 
the vibrations of associated matter to explain ordinary dispersion. 

I only found this forenoon that Loinim-l (Ann. der Ph. und 
Chem. 1878, vol. iv. p. 55) also goes on to double refraction of light 
in crystals the very problem I am breaking my head against. 
He is satisfied with his solution, but I do not think it at all satis- 
factory. It is the kind of thing that I have seen for a long time, 
but could not see that it was satisfactory ; and I do see reason for its 
not being satisfactory . He goes on from that and obtains an 
equation which would approximately give Huyglu-ns' surface. I 
have not had time to determine how far it may be correct, but I 
believe it must essentially differ from Huyghens' wave-surface to 
an extent comparable with that experimentally disproved by 
Stokes in his experimental disproof of Rankine's theory explaining 
optical aeolotropy by difference of inertia in different directions. 
The exceedingly close agreement of Huyghens' surface with the 
facts of the case which Stokes has found, absolutely cuts the 
ground from under a large number of very tempting modes of ex- 
plaining double refraction. 



LECTUEE VIII. 

TUESDAY, Oct. 7, 5 P.M. 

WE shall take some fundamental solutions for wave motion 
such as we have already had before us, only we shall consider them 
as now applicable to non-condensational distortional waves, instead 
of condensational waves. We can take our primary solution in the 



1 . 2-7T , Ik + n .. . 

form <j> = - sin (r ct), where c = A/ - if the 



wave is con- 



densational, and A/- if the wave is distortional. But for a dis- 
p 

tortional wave we must also have 8=0. 



In the first place, if our value of c is A / - , 



we know that 

satisfies p ~ = nrf<f>. [I want very much a name for that symbol 

V 2 (delta turned upside down). I do not know whether, Prof. 
Ball, -you have any name for it or not; your predecessor, Sir 
William Hamilton, used it a great deal, and I think perhaps you 
may know of a name for it.] The conditions to be fulfilled by the 
three components of displacement, (f, 17, of a distortional wave 
are, in the first place, 



and we must have besides 



Thus , 77, f must be three functions, each fulfilling the same 
equation. There is a fulfilment of this equation by functions < ; 
and as we have one solution, we can derive other solutions from 



ROTATIONAL OSCILLATION IN ORIGIN. 81 

that by differentiation. Let us see then, if we can derive three Molar. 
solutions from this value of <f> which shall fulfil the remaining con- 
dition. It is not my purpose here to go into an analytical investi- 
gation of solutions ; it is rather to show you solutions which are 
of fundamental interest. Without further preface then, I will 
put before you one, and another, and then I will interpret them 
both. 

Take for example the following, which obviously fulfils the 

equation ^ + ? + ^=0: 

dx dy dz 



In each case the distance-terms only of our solution are what we 
wish. Thus 

d<> . 2-rr z d 2?r 



Remark that in this solution the displacement at a distance from 
the source is perpendicular to the radius vector ; i.e. we have 



Before going further, it will be convenient to get the rotation. It 
is an exceedingly convenient way of finding the direction of vibra- 
tion in distortional displacements. The rotations about the axes of 
x, y, z will be : 



These rotations are proportional to -^ , ^ , ^ ; that is 
to say, besides an x component equal to , we have an r com- 

ponent equal to -j . We have a rotation around the radius vector 
T. L. 6 



82 LECTURE VIII. PART I. 

Molar. r, and a rotation around the axis of as, whose magnitudes are 
proportional to -^ and - . 

If you think out the nature of the thing, you will see that 
it is this : a globe, or a small body at the origin, set to oscillating 
rotationally about Ox as axis. You will have turning vibrations 
everywhere ; and the light will be everywhere polarized in planes 
through Ox. The vibrations will be everywhere perpendicular to 
the radial plane through Ox. 

In the first place we have (omitting the constant factor J 



=0, 17 = - -^ cos ry, 



Hence for (y = 0, z = 0,) the displacements are zero, or we have 
zero vibration in the axis of x. Everywhere else the displacements 
are not null and are perpendicular to Ox (since we always have 
= 0); and being perpendicular also to the radius vector, they are 
perpendicular to the radial plane through the axis of x. 

Let us consider the state of things in the plane yz. Suppose 
we have a small body here at the origin or centre of disturbance, 
and that it is made to turn forwards and backwards 
in this way (indicating a turning motion about an axis 
JJ perpendicular to the plane of the paper) in any given 
period. What is the result ? Waves will proceed out 
in all directions from the source, and the intersections of the wave 
fronts with the plane (yz} of the paper will be circles. We shall 
have vibrations perpendicular to the radius vector; of magnitude 

^ , which is the same in all directions. The rotation (molecu- 
X r 

lar rotation about the axis of a; or in the plane yz,) is 

A. 

There is therefore zero displacement where the rotation or the 
distortion is a maximum (positive or negative) : and vice versa, at 
a place of maximum displacement (positive or negative) there is 
zero rotation and zero distortion. The rotation is clearly equal 
to half the distortion in the plane yz by shearing (or differential 
motion) perpendicular to rx, of infinitesimal planes perpendicular 



REALIZED ILLUSTRATION IN .TELLY. 83 

to r. The result is polarized light consisting of vibrations in the Molar. 
plane yz and perpendicular to the radius vector, and therefore 
the plane of polarization is the radial plane through OX. 

Here we have a simple source of polarized light; it is the 
simplest form of polarization and the simplest source that we can 
have. Every possible light consists of sequences of light from 
simple sources. Is it probable that the shocks to which the par- 
ticles are subjected in the electric light, or in fire, or in any 
ordinary source of light, would give rise to a sequence of this kind ? 
No, at all events not much ; because there cannot be much ten- 
dency in these shocks or collisions, to produce rotatory oscillations 
of the gross molecules. We can arbitrarily do it, for we can do 
what we will with the particle. That privilege occurred to me 
in Philadelphia last week, and I showed the vibrations by having 
a large bowl of jelly made with a ball placed in the middle of it. 
I really think you will find it interesting enough to try it for 
yourselves. It allows you to see the vibrations we are speaking 
of. I wish I had it to show you just now, so that you might see 
the idea realized. It saves brain very much. 

I had a large glass bowl quite filled with yellowish transparent 
jelly, and a red-painted wooden l>all floating in the middle of it. 
Try it, and you will find it a very pretty il- 
lustration. Apply your hand to the ball, and 
give it a turning motion round its vertical 
diameter, and you have exactly the kind 
of motion expressed by our equations. The 
motion in any oblique direction, such as at 

this point P (x, y, z) represents that of polarized light vibrating 
perpend icularly to the radial plane (or plane through the vertical 
central axis). Tle amplitude of the vibration here (in the vertical 

axis) is zero ; here at the surface (in the plane yz} it is -cosq; and 
if you use polar coordinates, calling this angle (indicating on the 
diagram) then the amplitude here (at P} is - cos q sin 6, giving 
when 6 is a right angle the previous expression. 

I say that this is the simplest source and the simplest system of 
polarized light that we can imagine. But it can scarcely be induced 
naturally. The next simplest is a globe or-small body vibrating to- 

62 




84 LECTURE VIII. PART I. 

Molar. and-fro in one line. We will take the solution for that presently. 
Still we have not got up to the essential complexity of the 
vibration produced naturally by the simplest natural vibrator 
with unmoved centre of inertia. I may take my hand and, 
instead of the torsional oscillations which we have been con- 
sidering, I may give vertical oscillations to the globe in the 
jelly (and that makes a very pretty modification of the experi- 
ment), and people all call out, " 0, there is the natural time of 
the vibration, if you only leave the globe to itself," oscillating 
up and down in the jelly. But the case is not proper for an 
illustration of undulatory vibrations spreading out from a centre. 
We are troubled here also by reflection back, as it were, from the 
containing bowl, just as in experiments on a stretched rope to 
show waves running along it, we are troubled by the rope not 
being infinitely long. You can always see sets of vibrations 
running along the rope, and reflected back from the ends. But in 
this experiment with the jelly in the bowl, you do not see the 
waves travelling out at all because the distance to the boundary 
is not large enough in comparison with the wave-length ; and what 
you really see is a certain set of standing vibrations, depending on 
the finiteness of the bowl. But just imagine the bowl to be 
infinitely large, and that you commence making torsional oscilla- 
tions; what will take place? A spreading outwards of this kind 
of vibrations, the beginning being, as we shall see, abrupt. We 
shall scarcely reach that to-day, but we shall perhaps another day 
consider if the motion in the source begins and ends abruptly, the 
consequent abruptness* of the beginnings and endings of the vibra- 
tions throughout an elastic solid ; in every case in which the 
velocity of propagation is independent of the wave-length. 

When you apply your hands and force the ball to perform 
those torsional vibrations, you have waves proceeding from it ; 
but if you then leave it to itself, there is no vibrating energy in 
it at all, except the slight angular velocity that you leave it with. 
A vibrator which can send out a succession of impulses independ- 
ently of being forced to vibrate from without, must be a vibrator 
with the means of conversion of potential into kinetic energy in 
itself. A tuning-fork, and a bell, are sample vibrators for sound. 
The simplest sample vibrator that we can imagine to represent the 

* This is in fact proved by the solution of Lecture IV. expressed by (14) in terms 
of an arbitrary function. 



DOUBLE SOURCE WITH CONTRARY ROTATIONS. 85 

origin of ail independent sequence of light may be like a tuning-fork. Molar. 
Two bodies, joined by a spring would be more symmetrical than a 
tuning-fork. Two rigid globes joined by a spring that will give 
you the idea ; or (which will be a vibration of the same type still) 
one elastic spherical body vibrating from having been drawn into 
an oval shape, and let go. 

I will look, immediately, at a set of vibrations produced in an 
elastic solid by a sample vibrator. But suppose you produce 
vibrations in your jelly elastic solid by taking hold of this ball 
and moving it to-aud-fro horizontally, or again moving it up and 
down vertically and think of the kinds of vibrations it will make 
all around. Think of that, in connection with the formulas, and 
it will help us to interpret them. This is in fact the kind of 
vibration produced in the ether by the rigid spheric.il containing 
shell of our complex molecule (Lecture II.). Or think of the 
vibrations due to the higher though simpler order of vibrator, of 
which we have taken as an example a very dense elastic globe 
vibrating from prolate to oblate and back periodically. We might 
also have those torsional vibrations ; but among all the possible 
vibrations of atoms in the clang and clash of atoms that there is 
in a flame, or other source of light, a not very rare case I think 
would be that which I am going to speak of now. It consists of 
opposite torsional vibrations at the two ends of an elongated mass. 
To simplify our conception for a moment, imagine two globes con- 
nected by a columnar spring; twist them in opposite directions, and 
let them go. There you have an imaginable source of vibrations. 
If in any one of our cases the potential energy of the spring is very 
large in comparison with the energy that is carried off in a 
thousand, or a hundred thousand vibrations, you will have a 
nearly uniform sequence of vibrations such as those we have been 
considering, but gradually dying down. 

Before passing on to the to-and-fro vibrator we will think of 
this motion for a moment, but we will not work it out, because it 
is not so interesting. To suit our drawing we shall suppose one 
globe here, and another upon the opposite side on a level with the 
first, so that the line of the two is perpendicular to the board 
Give these globes opposite torsional vibrations about their common 
axis, and what will the result be ? A single one produces zero 
light in the axis and maximum light in the equatorial plane. The 
two going in opposite directions will produce zero light in the 



86 LECTURE VIII. PART I. 

Molar. equatorial plane and zero light in the axis ; so that you will 
proceed from zero in the equatorial plane to a maximum between 
the equatorial plane and the poles, and zero at the poles ; and you 
will have opposite vibrations in each hemisphere. That constitutes 
a possible case of vibrations of polarized light, proceeding from a 
possible independent vibrator. 

One of the most simple and natural suppositions in respect to 
an independent vibrator is afforded by the illustration of a bell, or 
a tuning-fork, or an elastic body deformed from its natural shape 
and left to vibrate. In all these cases, as also in the case of our 
supposed complex molecule (Lecture I.), remark that the centre 
of gravity of the vibrator is at rest ; except for the comparatively 
very small reaction of the ether upon it ; and this is essential to an 
independently acting vibrator. The vibrator must have potential 
energy in itself, for many thousand vibrations generating waves 
travelling outwards through ether ; and its centre of gravity must 
be at rest, except in so far as the reaction of the medium upon 
it causes a slight motion of the centre of gravity. 

I will put down the solution which corresponds to a to-and-fro 
vibration in the axis of x, viz. : 



</> is our old friend, 

1 . 27T/ AA 

- sin (r - t A / - , 
A. V V p/ 

but with n now in place of the n + </ which we had formerly when 
we were dealing with condensational waves. First remark that 
we know that 



are satisfied, because </> and all its differential coefficients satisfy 
this relation. We have therefore only to verify that the dilatation is 
zero. Instead of merely going through the verification, I wish to help 
you to make the solution your own by showing you how I obtained 
it. I will not say that there is anything novel in it, but it is simply 
the way it occurred to me. I obtained it to illustrate Stokes' 
explanation of the blue sky. I afterwards found that Lord Rayleigh 
had gone into the subject even more searchingly than Stokes, and 
I read his work upon it. 



WAVES OENERATED BY TO-AND-FRO VIBRATOR. 87 

The way I found the solution was this : is clearly the dis- Molar. 

placement-p>tential for an elastic fluid, corresponding to a source 
of the kind, constituted by an immersed solid moving to-and-fro 
along the axis of x. The displacement function of which the 
displacements are the differential coefficients would take simply that 
form if the question were of sound in air or other compressible 
fluid, and not of light, or of waves in an incompressible solid, or of 
waves of distortion in a compressible solid. It was a question of 
condensational vibrations with us several days ago. I did not go 
into the matter in detail then, but we saw that for condensational 
vibrations proceeding from a vibrator vibrating to-and-fro along 

the axis of x that , was the displacement potential ; and it is 
obvious, if we start from the very root of the matter that it must be so. 
Hence we may judge that the differential coefficients T_, -7-, -.- 

of j^ , with it in place of n -I- 1&, must therefore be at all events 

constituents of the components of displacement in the case of 
light, or of distortional waves, from such a source : but neither they 
nor the differential coefficients of any function can be simply 
equal to the displacement-components of our present problem, in 
which the motion is essentially rotational. The irrotational dis- 
placements in the condensational wave problem are displacements 
which fulfil certain of the conditions of our present problem : but 
they do not fulfil the condition of giving us a purely distortional 
wave, unless we add a term or terms in order to make the dilata- 
tion zero. This is done in fact, as I found, by the addition of the 

spherically symmetric term , < to -y- ^- , for the ^-component of 
displacement. J ust try for the dilatation. We have 



in which we may substitute -, for <f>. Thus 



_ 
dx ~ ~ X ' dx ' 



88 LECTURE VIII. PART I. 

Molar. We verify therefore in a moment that the displacements given by 
the formulas satisfy 

+ *. + *_ . 

dx dy dz 

and thus we have made up a solution which satisfies the con- 
ditions of being rigorously non-condensational, (no condensation 
or rarefaction anywhere,) and of being symmetrical round the axis 
of x. 

In the first place, taking the distant terms only, we have 



4?r 2 xy . 



It is easy to verify that these displacements are perpendicular to 
the radius vector, i.e. that we have x% + y^ + z= 0. Just look at 
the case along the axis of x, and again in the plane yz. It is 
written down here in mathematical words painting as clearly and 
completely as any non-mathematical words can give it. Take 
y = 0, z = 0, and that makes = 0, 17 = 0, f = 0. Therefore, in the 
direction of the axis of x there is no motion. That is a little 
startling at first, but is quite obviously a necessity of the funda- 
mental supposition. Cause a globe in an elastic solid to vibrate 
to-and-fro. At the very surface of the globe the points in which 
it is cut by Ox have the maximum motion ; and throughout the 
whole circumference of the globe, the medium is pulled, by hypo- 
thesis, along with the globe. But this is not a solution for that 
comparatively complex, though not difficult, problem. I am only 
asking you to think of this as the solution for the motion at a great 
distance. It may not be a globe, but a body of any shape moved to- 
and-fro. To think of a globe will be more symmetrical. In the im- 
mediate neighbourhood of the vibrator there is a motion produced in 
the line of vibration; the motion of the elastic solid in that neigh- 
bourhood consists in a somewhat complex, but very easily imagined 
state of things, in which we have particles in the axis of #, moving 
out and in directly along the radius vector; in all other places 
except the plane of yz, slipping around with motions oblique to the 
radius vector ; and in the plane of yz moving exactly perpendicular 



WAVES GENERATED BY TO-AND-FRO VIBRATOR. 89 

to the radius vector. All, however, except motions perpendicular Molar, 
to the radius vector, become insensible at distances very great in 
comparison with the wave-length. We have taken, simply, the 
leading terms of the solution. These represent the motion at 
great distances, quite irrespectively of the shape of the body, 
and of the comparatively complicated motion in the neighbourhood 
of the vibrating body. 

Take now x = 0, and think of the motions in the plane yz. 
The vibrator is supposed to be vibrating perpendicular to this 
plane. We have 

T7T 1 . -. 

= ^ r -sinq, rj = 0, f = 0. 

What does that mean ? Clearly, that the vibrations are perpen- 
dicular to the plane yz. We have the wave spreading out uni- 
formly in all directions in that plane, and " polarized in " that 
plane, the vibrations being perpendicular to it. That is exactly 
what Stokes supposed was of necessity the dynamical theory of 
the blue light of the sky. Lord Rayleigh showed that it was not 
so obvious as Stokes had supposed. He elaborately investigated 
the question, " Whether is the blue light of the sky, (which we 
assume to be owing to particles in the air,) due to the particles 
being of density different from the surrounding luminiferous ether, 
or being of rigidity different from the surrounding luminiferous 
ether?" The question would really be, If the particles are water, 
what is the theory of waves of light in water; does it differ from 
air in being, as it were, a denser medium with the same effective 
rigidity, or is it a medium of the same density and less effective 
rigidity, or does it differ from ether both as to density and as to 
rigidity ? 

Lord Rayleigh examined that question very thoroughly, and 
finds, if the fact that the cause were, for instance, little spherules 
of water, and if in the passage of light through water the propaga- 
tion is slower than in air were truly explained by less rigidity and 
the same density we should have something quite different in the 
polarization of the sky from what we would have on the other 
supposition. On the other hand, the observed polarization of the 
sky supports the other supposition (as far as the incertitude 
of the experimental data allows us to judge) that the particles, 
whether they be particles of water, or motes of dust, or whatever 



90 LECTURE VIII. PART I. 

they may be, act as if they were little portions of the luminiferous 
ether of greater density than, and not of different rigidity from, 
the surrounding ether. Hence our present solution, which has for 
us such special interest as being the expression of the disturbance 
produced in the ether by our imbedded spring-molecule, acquires 
farther and deeper interest as being the solution for dynamical 
action which according to Stokes and Rayleigh is the origin cause 
of the blue light coming from the sky. I will call attention a 
little more to Lord Rayleigh's dynamics of the blue sky in a 
subsequent lecture. Meantime, returning to our solution, we 
may differentiate once more with respect to x, in order to get 
a proper form of function to express the motion from a double 
vibrator vibrating to-and-fro like this vibrating my hands to- 
wards and from each other. Then we shall have a solution which 
will express another important species of single sequence of vibra- 
tions, of which multitudes may constitute the whole, or a large 
part, of the light of any ordinary source. 

A question is now forced upon us, what is the velocity of a 
group of waves in the luminiferous ether disturbed by ordinary 
matter ? With a constant velocity of propagation, as in pure ether, 
each group remains unchanged. But how about the propagation 
of light-sequences in a transparent medium like glass ? It is a 
question that is more easily put than answered. We are bound 
to consider it most carefully. I do not despair of seeing the 
answer. I think, if we have a little more patience with our 
dynamical problem we shall see something towards the answer. 

Here is a perfectly parallel problem. Commence suddenly to 
give a simple harmonic motion through the handle P to our 
system of particles m iy in 2 ,...m j , which play the part of a molecule. 
If you commence suddenly imparting to the handle a motion of 
any period whatever, only avoiding every one of the fundamental 
periods, if there be a little viscosity it will settle into a state of 
things in which you have perfectly regular simple harmonic vibra- 
tion. But if there be no viscosity whatever, what will the result 
be? It will be composed of simple harmonic motions in the 
period of our applied motion at the bell-handle P; with the ampli- 
tude of each calculated from our continued fraction ; and super- 
imposed upon it, a jangle as it were, consisting of coexistent simple 
harmonic vibrations of all the fundamental periods. If there is no 
viscosity, that state of things will go on for ever. I cannot satisfy 



SUDDEN AND GRADUAL COMMENCEMENTS OF VIBRATION. 91 

myself with viscous terras in these theories not only because the Molecula 
assumption of viscosity, i molecular dynamics, is a theoretic 
violation of the conservation of energy ; but because the smallest 
degree of viscosity of ether sufficing to practically rid us of any 
of this jangling, or to have any sensible influence in any of the 
motions we have to do with in sources or waves of light, would not 
allow a light-sequence through ether to last, as we know it lasts, 
through millions of millions of millions of millions of vibrations. 
But if we have no viscosity at all, whatever energy of any vibra- 
tions, regular or irregular, we have at any time in our complex 
molecule must show in the vibrations of something else, and 
that is what ? In studying that sort of vibration with which we 
have been occupied in the molecular part of our course, we must 
account for these irregular vibrations somehow or other. The 
viscous terms which Helmholtz and others have introduced re- 
present merely an integral effect, as it were, of actions not followed 
in detail, not even explained, in the theory. By viscous terms, I 
mean terms that assume a resistance in simple proportion to 
velocity. 

But the state of things with us is that that jangling will go on 
for ever, if there is no loss of energy ; and we want to coax our 
system of vibrators into a state of vibration with an arbitrarily 
chosen period without viscous consumption of energy. Begin 
thus : commence suddenly acting on P just as we have already 
supposed, but with only a very small range of motion of P. The 
result will be just as I have said, only with very small ranges 
of all the constituent motions. After waiting a little time increase 
the range of the motion of P ; after waiting a little longer, in- 
crease the range farther, and so go on, increasing the range by 
successive steps. Each of those will superimpose another state of 
vibration. There would be, I believe, virtually an addition of the 
energies, not of the amplitudes, of the several jang lings if you make 
these steps quite independent of one another. 

For example, suppose you proceed thus: In the first place, 
start right off into vibrations of your handle P through a space, 
say of 30 inches. You will have a certain amount of energy, J, 
in the irregular vibrations (the "jangling"). In the second place, 
commence with a range of three inches. After you have kept 
P vibrating three inches through many periods, suddenly increase 
its range by three inches more, making it six inches. Then, some- 



92 LECTURE VIII. PART I. 

Molecular, time after, suddenly increase the range to nine inches ; and so on 
in that way by ten steps. The energy of the jangle produced by 
suddenly commencing through the range of three inches, which 
is one-tenth of 30 inches, will be exactly one-hundredth of J, the 
energy of jangle which you would have if you commenced right 
away with the vibration through 30 inches. Each successive 
addition of three inches to the range of P will add an amount 
of energy of jangling of which the most probable value is the one- 
hundredth of J; and the result is that if you advance by these 
steps to the range of 30 inches, you will have in the final jangle 
ten-hundredths, that is to say one-tenth, of the energy of jangle 
which you would get if you began at that range right away. 
Thus, by very gradually increasing the range, the result will be 
that, without any viscosity at all there will be infinitely little of 
the irregular vibrations. 

But there are cases in which we have that tremendous jangling 
of the molecules concerned in luminous vibrations ; for instance, 
the fluorescence of such a thing as uranium glass or sulphate 
of quinine which lasts for several thousandths of a second after 
the exciting light is taken away, and then again in phosphorescence 
that lasts for hours and days. There have been exceedingly 
interesting beginnings, in the way of experiments already made, 
in these subjects, but nobody has found whether initial refraction 
is exactly the same as permanent refraction. For this purpose 
we might use Becquerel's phosphoroscope or we might use methods 
such as those of Fizeau or Foucault, or take such an appliance as 
Prof. Michelson has been recently using, for finding the velocity of 
light, and so get something enormously more searching than even 
Becquerel's phosphoroscope, and try whether in the first hundredth 
of a second, or the first millionth of a second, there is any 
indication of a different wave velocity from that which we find 
from the law of refraction, when light passes continuously through 
a transparent liquid or solid. If, with the methods employed for 
ascertaining the velocity of light in a transparent body (to take 
account of the criticisms that they have received at the British 
Association meeting, to which I have referred several times), we 
combine a test for instantaneous refraction, it seems likely that 
we should not get negative results, but rather find phenomena 
and properties of ultimate importance. We might take not only 
ordinary transparent solids aud liquids, but also bodies in which, 



FLUORESCENCE AND PHOSPHORESCENCE. 93 

like uranium glass, the phosphorescence lasts only a few Molecular 
thousandths of a second ; and then again bodies in which phos- 
phorescence lasts for minutes and hours. With some of those 
we should have anomalous dispersion, gradually fading away after 
a time. I cannot but think that by experimenting, in some such 
way, we should find some very interesting and instructive results 
in the way of initial fluorescence. 



LECTURE IX. 

WEDNESDAY, Oct. 8, 5 P.M. 

WE shall go on for the present with the subject of the 
propagation of waves from a centre. Let us pass to the case of 
two bodies vibrating in opposite directions, by superposition of 
solutions such as that which we have already found for a single 
to-and-fro vibrator, which was expressed by 

47T 2 defy dd dd<> 



We verified that 



so that this expresses rigorously a distortional wave. It is obvious 
that this expresses the result of a to-and-fro motion through the 
origin in the line OA'. Remark, for one thing, that in the 
neighbourhood of the origin, at such moderate distance from it 
that the component motion in the direction OX is not insensible, 
we have on the two sides of the origin simultaneously positive 
values. is the same for a positive value of ./: as for the negative 
of that value. At distances from the origin in the line OX which 
are considerable in comparison with the wave-length the motion 
vanishes as we have seen. 

Pass on, now, to this case: a positive to-and-fro motion on the 
one side of the origin, and a simultaneous negative to-and-fro 
motion on the other side of the origin ; that is to say, two simul- 
taneous co-periodic vibrations of portions of matter on the two 
sides of the origin moving simultaneously in opposite directions. 
I will indicate these motions by arrow-heads, continuous arrow- 
heads to indicate directions of motion at one instant of the period, 



CONTRARY VIBRATORS IN ONE LINE. 



95 



and dotted at the instant half a period earlier or later. The first Molar, 
case already considered 

Fig.i. 

y 



the second case 



Fig. 2. 



.v. 



The effect in the first case being expressed by the displacements 
, 17, f, already given, the effect in the second case will be 
expressed by displacement-coefficients respectively equal to 

d dr, d$ 

dx' dx' dx' 

This configuration of displacement clearly implies a motion of 
which the component parallel to OX has opposite signs, and the 
components perpendicular to OX are equal with the same sign, for 
equal positive and negative values >}'.>; it is a simultaneous out 
and in vibration on the two sides of the origin in the line OX, and 
a simultaneous in and out vibration perpendicular to OX, every- 
where in the plane yz. A motion of the matter at distances from 
the origin moderate in comparison with the wave-length will l>e 
accurately expressed by these functions. Passing now from Fig. 2 
which shows the germ from which we have developed the idea of 
this configuration of motion, and the functions expressing it ; look 

Fig. 8. 
Y 




96 LECTURE IX. PART I. 

to Fig. 3 illustrating the in and out vibration perpendicular to OX 
which accompanies the out and in vibration along OX from which 
we started. The configuration of arrow-heads on the circle in 
Fig. 3 shows the component motions perpendicular to the radius 
vector at any distance, small or great, from the origin ; which 
constitute sensibly the whole motion at the great distances. To 
express this motion, take only the " distance-terms " (as in previous 

Q_3 

cases,) and drop the factor 3 , from the differential coefficients 
A, 

indicated above. We thus find, for the three components of the 
displacement at great distances from the origin, 

2 -r 2 ) . aty x 2 z 

f = tf v ^ 'cosq, r)=^cosq, -^ T cos?. 

To satisfy ourselves that the radial component of the displace- 
ment is zero verify that we have x% + yij + z% 0. 

To think of the kind of " polarization " that will be found, when 
the case is realized in a sequence of waves of light, remark that the 
motion is everywhere symmetrical around the axis of .r, and is in 
the radial plane through OX. Therefore, we have light polarized 
in the plane through the radius to the point considered and 
perpendicular to the plane through OX. 

This is, next to the effect of a single to-and-fro rigid vibrator, 
the simplest set of vibrations that we can consider as proceeding 
from any natural source of light. As I said, we might conceive of 
a pair of equal and opposite torsional motions at the two ends of 
a vibrating molecule. That is one of the possibilities, and it 
would be rash to say that any one possible kind of motion does not 
exist in so remarkably complex a thing as the motion of the 
particles from which light originates. 

The motion we have just now investigated is perhaps the most 
interesting, as it is obviously the simplest kind of motion that can 
proceed from a single independent non-rotating vibrator with 
unmoved centre of inertia. If you consider the two ends of a 
tuning-fork, neglecting the prongs, so that everything may be 
symmetrical around the two moving bodies, you have a way by 
which the motion may be produced. Or our source might be 
two balls connected by a spring and pulled asunder and set to 
vibrating in and out ; or it might be an elastic sphere which has 
experienced a shock. An infinite number of modes of vibration 
are generated when an elastic ball is struck a blow, but the gravest 



SUDDEN BEGINNINGS AND GRADUAL SURSIDENCES. 97 

mode, which is also no doubt the one in which the energy is Molar, 
greatest, if the impinging body be not too hard, consists of the 
globe vibrating from an oblate to a prolate figure of revolution ; 
and this originates in the ether the motion with which we have 
just now l>een occupied. 

The kind of thing that an elemental source of light in nature 
consists of, seems to me to be a sudden initiation of a set of 
vibrations and a sequence of vibrations from that initiation which 
will naturally become of smaller and smaller amplitude. So that 
the graphic representation of what we should see if we could see 
what proceeds from one element of the source, the very simplest 
conceivable element of the source, would consist of polarized waves 
of light spreading out in all directions according to some such law 
as we have here. In any one direction, what will it be ? Suppose 
that the wave advances from left to right ; you will then see what 
is here represented on a magnified scale. 



I have tried to represent a sudden start, and a gradual falling 
off of intensity. Why a sudden start ? Because I believe that 
the light of the natural flame or of the arc-light, or of any other 
known source of light, must be the result of sudden shocks upon 
a number of vibrators. Take the light obtained by striking two 
quartz pebbles together. You have all seen that. There is one 
of the very simplest sources of light Some sort of a chemical or 
ozoniferous effect connected with it which makes a smell, there must 
be. As to what the cause of this peculiar smell may be, I suppose 
we are almost assured, now, that it proceeds from the generation 
of ozone. What sort of a thing can the light be that proceeds 
from striking two quartz pebbles together ? Under what cir- 
cumstances can we conceive a group of waves of light to begin 
gradually and to end gradually ? You know what takes place in 
the excitation of a violin string or a tuning fork by a bow. The 
vibrations gradually get up from zero to a maximum and then, 
when you take the bow off, gradually subside. I cannot 
see anything like that in the source of light. On the contrary, 
it seems to me to be all shocks, sudden beginnings and gradual 
subsidences; rather like the excitation of a harp string plucked 
in the usual manner, or of a pianoforte string struck by the 
T.L. 7 



98 LECTURE IX. PART I. 

hammer ; and left to itself to give away all its energy gradually 
in waves of sound. 

I say this, because I have just been reading very interesting 
papers by Lommel and Sellineier*, both touching upon this subject. 
Helmholtz remarks that Sellmeier gets into a difficulty in his 
dynamics and does not show clearly what becomes of the energy 
in a certain case ; but it seems to me that Sellmeier really takes 
hold of the thing with great power. He goes into this case 
very fully, and in a way with which we are all now more or 
less familiar. He remarks that Fizeau obtained a suite of 50,000 
vibrations interfering with one another, and judges from that that, 
though ordinary light consists of polarized light, circularly-, or ellipti- 
cally-, or plane-polarized as I said to you myself, one or two days ago, 
with (what I did not say) the plane of polarization, or one or both 
axes of the ellipse if it be elliptically polarized, gradually varying, 
and the amplitude gradually changing, the changing must be 
so gradual that the whole amount of the change, whether of 
amplitude or of mode of polarization or of phase, in the course 
of 50,000 or 100,000, or perhaps several million vibrations cannot 
be so great as to prevent interference. In fact, I suppose there is 
no perceptible difference between the perfectness of the annul- 
ments in Fizeau's experiment, with 50,000 vibrations and with 
1,000 ; although I speak here not with confidence and I may be 
corrected. You have seen that with your grating, have you not, 
Prof. Rowland ? 

PROF. ROWLAND. Yes; but it is very difficult to get the 
interferences. 

SIR WM. THOMSON. But when you do get them, the black 
lines are very black, are they not ? 

PROF. ROWLAND. I do not know. They are so very faint 
that you can hardly see them. 

SIR WM. THOMSON. What do you infer from that ? 

PROF. ROWLAND. That there is a large number. The narrow- 
ness of the lines of the spectrum indicates how perfectly the light 
interferes ; and with a grating of very fine lines I find exceedingly 
perfect interference for at least 100,000 periods I should think. 

SIR WM. THOMSON. That goes further than Fizeau. Sellmeier 
says that probably a great many times 50,000 waves must pass 
before there can be any great change. He goes at the thing very 

* Kellmeier; Ann. dcr Phij. u. Chem. 1872, Vols. CXLV., CXLVII. 



AMOUNT OP ENEROY IN THE VIBRATION OF AN ATOM. 99 

admirably for the foundation of his dynamical explanation of Molar, 
absorption and anomalous refraction. The only thing that I do 
not fully agree with him in his fundamentals is the gradualness of 
the initiation of light at the source. I believe, in the majority of 
cases at all events, in sudden beginnings and gradual endings. 
Prof. Rowland has just told us how gradual the endings are. 
Fizeau could infer that the amplitude does not fall off greatly in 
50,000 vibrations. It is quite possible from all we know, that the 
amplitude may fall off considerably in 100,000 vibrations, is 
it not ? 

PROF. ROWLAND. The lines are then very sharp. 
SIR WM. THOMSON. It would not depend on the sharpness of 
the lines, would it ? 

PROF. ROWLAND. O, yes. It would draw them out of 
fineness. 

SIR WM. THOMSON. Would it broaden them out, or would 
it leave them fine, but throw a little light over a place that should 
be dark? 

PROF. ROWLAND. It would broaden them out. 
SIR WM. THOMSON. It is a very interesting subject; and 
from the things that have been done by Prof. Rowland and others, 
we may hope to see, if we live, a conquering of the difficulties 
quite incomparably superior to what we have now. I have no 
doubt, however, but that some now present will live to see 
knowledge that we can have hardly any conception of now, of the 
way of the extinction of vibrations in connection with the origin 
and the propagation of light. We are perfectly certain that the 
diminution of amplitude in the majority of sequences in any 
ordinary source, must be exceedingly small practically nil in 
1 ,000 vibrations ; "/e can say that probably it is practically nil 
in 50,000 vibrations: we know that it is nearly nil in 100,000 
vibrations. Is it practically nil in two or three hundred thousand 
vibrations, or in several million vibrations ? Possibly not. Dy- 
namical considerations come into play here. We shall be able 
to get a little insight into these things by forming some sort of an 
idea of the total amount of energy there can possibly be in one 
elemental vibrator, in a source of light, and what sequences of 
waves it can supply. That the whole energy of vibration of 
a single freshly excited vibrator in a source of light is many 
times greater than what it parts with in the course of 100,000 

7-2 



100 LECTURE IX. PART I. 

Molar. molar-vibrations, is a most interesting experimental conclusion, 
drawn from Fizeau's and Rowland's grand observations of inter- 
ference. 

In speaking of Sellmeier's work, and Helmholtz's beautiful 
paper which is really quite a mathematical gem, I must still say 
that I think Helmholtz's modification is rather a retrograde step. 
It is not so perhaps in the mathematical treatment; and at the 
same time Helmholtz is perfectly aware of the kind of thing that 
is meant by viscous consumption of energy. He knows perfectly 
well that that means, conversion of energy into heat ; and in 
introducing viscosity he is throwing up the sponge, as it were, 
so far as the fight with the dynamical problem is concerned. 

Mr Mansfield brought me another quarter hundred weight of 
books on the subject last night. I have not read them all 
through. I opened one of them this forenoon, and exercised 
myself over a long mathematical paper. I do not think it will 
help us very much in the mathematics of the subject. What we 
want is to try and see if we cannot understand more fully what 
Sellmeier has done, and what Lommel has done. I see that both 
stick firmly to the idea that we must in the particles themselves 
account for the loss of energy from the transmitted wave. That is 
what I am doing; and we shall never have done with it until we 
have explained every line in Prof. Rowland's splendid spectrum. 
If we are tired of it, we can rest, and go at it again. 

Lommell and Sellmeier do not go very fully into these 
multiple modes of vibrations, although they take notice of them. 
But they do indicate that we must find some way of distributing 
the energy without supposing annulment of it. That is the 
reason why I do not like the introducing of viscous terms in our 
equations. It is very dangerous, in an ideal sense, to introduce 
them at all. This little bit of viscosity in one part of the system 
might run away with all our energies long before 50,000 vibrations 
could be completed. If there were any sensibly effective viscosity 
in any of the material connected with the moving particle it might 
be impossible to get a sequence of one-hundred thousand or a million 
vibrations proceeding from one initial vibration of one vibrator. 

What the dynamical problem has to do for us is to show how 
we can have a system capable of vibrations in itself and acted upon 
by the luminiferous ether, that under ordinary circumstances does 
not absorb the light in millions of vibrations, as for transparent 



DYNAMICS OK ABSORPTION FIRST TAUGHT BY STOKES. 101 

liquids or solids, or in hundreds of thousands of millions of vibra- Molar, 
tions as in our terrestrial atmosphere. That is the case with 
transparent bodies ; bodies that allow waves to pass through them 
one-hundred feet or fifty miles, or greater distances ; transparent 
bodies with exceedingly little absorption. If we take vibrators, 
then, that will perform their functions in such a way as to give a 
proper velocity of propagation for light in a highly transparent 
body, and yet which, with a proper modification of the magnitudes 
of the masses or of the connecting springs, will, in certain complex 
molecules, such as the molecules of some of those compounds that 
give rise to fluorescence and phosphorescence, take up a large 
quantity of the energy, so that perhaps the whole suite of 
vibrations from a single initiation may be absolutely absorbed 
and converted into vibrations of a much lower period, which will 
have, lastly, the effect of heating the body, I think we shall see 
a perfectly clear explanation of absorption without introducing 
viscous terms at all; and that idea we owe to Sellmeier. 

I would like, in connection with the idea of explaining 
absorption and refraction, and lastly, anomalous refraction and 
dispersion, to just point out as a matter of history, the two 
names to which this is owing, Stokes and Sellmeier. I would 
l>e glad to be corrected with reference to either, if there is any 
evidence to the contrary; but so far as I am aware, the very first 
idea of accounting for absorption by vibrating particles taking up, 
in their own modes of natural vibration, all the energy of those 
constituents of mixed light trying to pass through, which have 
the same periods as those modes, was from Stokes. He taught 
it to me at a time that I can fix in one way indisputably. 
I never was at Cambridge once from about June 1852 to May 
1865; and it was at Cambridge walking about in the grounds 
of the colleges that I learned it from Stokes. Something was 
published of it from a letter of mine to Helmholtz, which he 
communicated to Kirchhoff and which was appended by Kirch hoff 
in his postscript to the English translation (published in Phil. 
Mag., July 1860) of his paper on the subject which appeared in 
Poggendorff's Annalen, Vol. CIX. p. 275. 

In the postscript you will find the following statement taken 
from my letter: 

" Prof. Stokes mentioned to me at Cambridge some time ago, 
probably about ten years, that Prof. Miller had made an experiment 



102 LECTURE IX. PART I. 

Molar. testing to a very high degree of accuracy the agreement of the 
double dark line D of the solar spectrum, with the double bright 
line constituting the spectrum of the spirit lamp with salt. I 
remarked that there must be some physical connection between 
two agencies presenting so marked a characteristic in common. 
He assented, and said he believed a mechanical explanation of 
the cause was to be had on some such principle as the following: 
Vapour of sodium must possess by its molecular structure a ten- 
dency to vibrate in the periods corresponding to the degrees of 
refrangibility of the double line D. Hence the presence of sodium 
in a source of light must tend to originate light of that quality. 
On the other hand, vapour of sodium in an atmosphere round a 
source must have a great tendency to retain in itself, i.e. to absorb, 
and to have its temperature raised by, light from the source, of 
the precise quality in question. In the atmosphere around the 
sun, therefore, there must be present vapour of sodium, which, 
according to the mechanical explanation thus suggested, being 
particularly opaque for light of that quality prevents such of it as 
is emitted from the sun from penetrating to any considerable 
distance through the surrounding atmosphere. The test of this 
theory must be had in ascertaining whether or not vapour of 
sodium has the special absorbing power anticipated. I have the 
impression that some Frenchman did make this out by experiment, 
but I can find no reference on the point. 

" I am not sure whether Prof. Stokes' suggestion of a me- 
chanical theory has ever appeared in print. I have given it in 
my lectures regularly for many years, always pointing out along 
with it that solar and stellar chemistry were to be studied by 
investigating terrestrial substances giving bright lines in the 
spectra of artificial flames corresponding to the dark lines of the 
solar and stellar spectra*." 

* [The following is a note appended by Prof. Stokes to his translation of a paper 
by Kirchhoff in Phil. Mag., March 1860, p. 190: "The remarkable phenomenon 
discovered by Foucault, and rediscovered and extended by Kirchhoff, that a body 
may be at the same time a source of light giving out rays of a definite refrangibility, 
and an absorbing medium extinguishing rays of the same refrangibility which 
traverse it, seems readily to admit of a dynamical illustration borrowed from sound. 
We know that a stretched string which on being struck gives out a certain note 
(suppose its fundamental note) is capable of being thrown into the same state of 
vibration by aerial vibrations corresponding to the same note. Suppose now a por- 
tion of space to contain a great number of such stretched strings forming thus the 



SELLMKIEK'S DYNAMICS OF ANOMALOUS DISPERSION. 103 

What I have read thus far is with reference not to the Molar, 
origin of spectrum analysis, but to the definite point, of Stokes' 
suggested dynamics of absorption. There is no hint there of the 
effect of the reaction of the vibrating particles in the luminiferous 
ether in the way of affecting the velocity of the propagation of 
the light through it. Sellmeier's first title has reference to that 
effect ; he explains ordinary refraction through the inertia of 
these particles and he shows how, when the light is nearly of the 
period corresponding to any of the fundamental periods of the 
embedded vibrators, there will be anomalous dispersion. He gives 
a mathematical investigation of the subject, not altogether satis- 
factory, perhaps, but still it seems to me to formulate a most 
valuable step towards a wholly satisfactory treatment of the thing. 
Lord Rayleigh, Helmholtz and others have quoted Sellmeier. 
Lommcl begins afresh, I think, but he notices Sellmeier also, so 
the idea must have originated with Sellmeier, and it seems to me 
a very important new departure with respect to the dynamical 
explanation of light. 

Ten minutes interval. 

Now, let us look at this problem of vibrating particles once Moleeul* 
more. I have a little exercise to propose for the ideal arithmetical 
laboratory. Just try the arithmetical work for this problem for 
7 particles. I do not know whether it will work out well or not. 
I have not the time to do it myself, but perhaps some of you may 
find the time, and be interested enough in the thing, to do it. 
Take the m's in order, proceeding by ratios of 4 ; and the c's in 
order, proceeding by differences of 1 : 

m,, m t , HI,, m 4 , w 6 , m a , W T = 1, 4, 16, 64, 256, 1024, 4096, 
c,. c t , c,, c 4 , c,, c -f c t , c, = 1, 2, 3, 4, 5, 6, 7, 8. 

There will be 7 roots to find by trial. I would like to have 
some of you try to find some of these, if not all ; also the energy 
ratios. You will probably find it an advantage in the calculation 

if you proceed thus : put - * = z, and by " roots " let us understand 

analogue of* "medium." It is evident that such a medium, on being agitated, 
would give out the note above mentioned, while on the other hand, if that note were 
sounded in air at a distance, the incident vibrations would throw the strings into 
vibration and consequently would themselves be gradually extinguished since other- 
wise there would be a creation of vis viva. The optical application of thi- illtmtia- 
tion is too obvious to need comment. G. G. 8." H.] 



104 LECTURE IX. PART II. 

Molecular, values of z, making = 0, which implies, and is secured by, u t = 0. 
We have 

a l = z 3, a a = 4s;-5, a s = I6z-7, a t = 64>z - 9, 
a, = 256.z -11, a e = 1024s -13, 7 = 4096.2 - 15. 

You will have to take values of z by trial until you get near a 
root. The convergence of the continued fraction (p. 39) will be 
so rapid that you will have very little trouble in getting the 
largest roots. Begin then with the largest -root, corresponding 
to the shortest of the critical periods r, and proceed downwards, 
according to the indications of pp. 55 58. In the course of 
the process, you will have the whole series of the its for each 
root ; by multiplying these in order, you have the a/'s for each par- 
ticular root, and then you can calculate the energy ratios for each 
root. We shall then be able to put our formula into numbers ; and 
I feel that I understand it much better when I have an example 
of it in numbers than when it is merely in a symbolic form. 

I want to show you now the explanation of ordinary refraction. 
Let us go back to our supposition of spherical shells, or, if you like, 
our rude mechanical model. Suppose an enormous number of 
spherical cavities distributed equally through the space we are con- 
cerned with. Let the quantity of ether thus displaced be so 
exceedingly small in proportion to the whole volume that the 
elastic action of the residue will not be essentially altered by that. 
These suppositions are perfectly natural. Now, what is unnatural 
mechanically, is, that we suppose a massless rigid spherical lining to 
this spherical cavity in the luminifcrous ether connected with an 
interior rigid massive shell, /// p by springs in the first place 
symmetrical. We shall try afterwards to see if we cannot do some- 
thing in the way of aeolotropy ; but as I have said before I do not 
see the way out of the difficulties yet. In the meantime, let us 
/^~1P~~"\ suppose this first shell m to 

Massless rigid slicll lining) / ^ ' \ . . 11 i i 

to spherical cavity in the! ./ /, -^ *v \ be isotropically connected by 

luminiferous ether. | / /^^TN^\ \ . . 

riA/ IAAAI^I VIA) s P nn e s Wlt 'k tne "&id s ^ e ^ 

sheii NO. i, mi l.Kl x. J / / lining of the spherical cavity 

\^^V/ in the ether. When I say 
^^ZllS-^ isotropically connected I 

mean distinctly this: that if you draw this first shell w, aside 
through a certain distance in any direction, and hold it so, the 



DYNAMICAL EXPLANATION OF ORDINARY REFRACTION. 105 

required force will be independent of the direction of the displace- Molecular, 
ment. Certain springs in the drawing the smallest number 
would be three placed around in proper positions will rudely 
represent the proper connections for us. Similarly, let there 
be another shell here, m r in the interior of m lt isotropically 
connected with it by springs ; and so on. 

This is the simplest mechanical representation we can give of a 
molecule or an atom, imbedded in the luminiferous ether, unless 
we suppose the atom to be absolutely hard, which is out of the 
question. If we pass frum this problem to a problem in which we 
shall have continuous elastic denser matter instead of a series of 
connections of associated particles, we shall be, of course, much 
nearer the reality. But the consideration of a group of particles 
has great advantage, for we are more familiar with common algebra 
than with the treatment of partial differential equations of the 
second order with coefficients not constant, but functions of the 
independent variable, which are the equations we have to deal 
with if we take a continuous elastic molecule, instead of one made 
up of masses connected by springs as we have been supposing. 

Let us suppose the diameters of these spherical cavities to be 
exceedingly small in comparison with the wave length. Practically 
speaking, we suppose our structure to be infinitely fine-grained. 
That will not in the least degree prevent its doing what we want. 
The distance also from one such cavity, containing within it a 
series of shells, to another such cavity, in the luminiferous ether, is 
to be exceedingly small in comparison with the wave length, so 
that the distribution of these molecules through the ether leaves 
us with a body which is homogeneous when viewed on so coarse a 
scale as the wave length ; but it is, if you like, heterogeneous 
when viewed with a microscope that will show us the millionth or 
million-millionth nf a wave length. This idea has a great advan- 
tage over Cauchy's old method, in allowing an infinitely fine- 
grainedness of the structure, instead of being forced to suppose 
that there are only several molecules, ten or twelve, to the wave 
length, as we are obliged to do in getting the explanation of 
refraction by Cauchy's method 

I wish to show you the effect of molecules of the kind now 
assumed upon the velocity of light passing through the medium. 

Let -r- denote the sum of all the masses of shells No. 1 in any 



106 LECTURE IX. PART II. 

Molecular, volume divided by the volume; let j-^ denote the sum of the masses 

of No. 2 interior shell in any volume divided by the volume ; and 
so on. We will not put down the equations of motion for all 
directions, but simply take the equations corresponding to a set of 
plane waves in which the direction of the vibration is parallel to 
OX, and the direction of the propagation is parallel to OF. 

If we denote by ~ z the density of the vibrating medium, (I am 

taking ^- 2 instead of the usual p for the reason you know, viz. : to 
get rid of the factor 4?r 2 resulting from differentiation). Let 
(instead of n as formerly,) denote the rigidity of the 

luminiferous ether. The dynamical equation of motion of the ether 
and embedded cavity-linings will clearly be 

' 



For waves of period T, we have = const. X sin 2?r ( 

\A, _/ 

The second differential coefficients of this with respect to t and x 
will be - - T2 , - - x - respectively. Therefore our equation be- 

comes -fL = 2 + c, (1 - -jH. Let us find ^ which is the reciprocal 

-I A, \ g / A, 

of the square of the velocity of propagation. You may write it 2 
if you like, or /u,' 2 , the square of the refractive index. We have, 



Substitute our value (Lecture VII.) for -a\l%; 



and this becomes 



T 2 i r f Cl r / ^/^ /c 2 2 

V=T[P-^ i+ ; n _r+, 



This is the expression for the square of the refractive index as 
it is affected by the presence of molecules arranged in the way we 



DYNAMICAL EXPLANATION OF ORDINARY REFRACTION. 107 

have supposed. It is too late to go into this for interpretation Molecular, 
just now, but, I will tell you that if you take T considerably less 
than *,, and very much greater than *,, you will get a formula 
with enough of disposable constants to represent the index of 
refraction by an empirical formula, as it were; which, from what we 
know, and what Sellmeier and Ketteler have shown, we can accept 
as ample for representing the refractive index of ordinary trans- 
parent substances. 

We shall look into this farther, a little later, and I will point 
out the applications to anomalous dispersion. We must think a 
good deal of what can become of vibrations in a system of that 
kind when the period of the vibration of the luminiferous ether 
is approximately equal to any one of the fundamental periods that 
the internal complex molecule could have were the shell lining in 
the ether held absolutely at rest. 



LECTURE X. 

THURSDAY, October 9, 5 P.M. 

Molar. WE shall now think a little about the propagation of waves 

with a view to the question, what is the result as regards waves 
at a distance from the source, the source itself being discontinuous 
in its action. In the first place, we will take our expression for a 
plane wave. The factor in our formulas showing diminution of 
amplitude at a distance from a source does not have effect 
when we come to consider plane waves. So we just take the 
simple expression for plane harmonic waves propagated along the 
axis of y with velocity v ; 

= a cos (y vt). 

A. 

Let us consider this question: what is the work done per 
period by the elastic force in any plane perpendicular to the line 
of propagation of the wave. We shall think of the answer to 
that question with the view to the consideration of the possibility 
of a series of waves advancing through space previously quiescent. 
Suppose I draw a straight line here for the line of propagation 
and let this curve represent a succession of waves travelling from 
left to right and penetrating into an elastic solid previously 



quiescent. Take a plane perpendicular to the line of propa- 
gation of the waves, and think of the work done by the elastic 
solid upon one side of this plane upon the elastic solid on the 



ENERGY OF WAVES. 109 

other side, in the course of a period in the vibration. We shall Molar, 
take an expression for the tangential force in the plane XOZ, 
and in the direction OX, which we denote by T (according 
to our old notation of S, T, U, P, Q, R). We shall virtually in- 
vestigate here the formula for the propagation of the wave in- 
dependently of our general formula in three dimensions. Taking 
T to denote the tangential force of the elastic medium on the one 
side of the plane XOZ, the downward direction of the arrow-head 
which I draw being that direction in which the medium on the 
left pulls the medium on the right, 1 put infinitely near that in 
the medium on the right another arrow-head. Imagine for the 
moment a split in the medium to indicate the reaction which the 
medium on the right exerts on the medium on the left by this 
plane ; and imagine the medium on the left taken away, and that 
you act upon the plane boundary of the medium on the right, 
with the same force as in the continuous propagation of waves. 
The medium upon the left acts in this way upon the plane inter- 
face : that is an easy enough conception. I correctly represent 
it in my diagram by an arrow-head pointing down infinitely near 
to the plane on the left-hand side. The displacement of the 
medium is determined by a distortion from a square figure to an 
oblique figure, and there is no inconsistency in putting into this 

little diagram an exaggeration of the obliquity, so as to 
| show the direction of it. The force required to do that 

is clearly as our diagram lies, upward on the right and 

downward on the left. 

Let us consider now the work done by that force. Calling 
the displacement of a particle from its mean position, T. f is 

the work done by that tangential force per unit of time. ~ is 

dy 

the shearing strain experienced in the medium so that 

n <| = ~ r 

In this particular position which we have taken, increases with y 
so that the sign minus is correct according to the arrow heads. 

Let there be simple harmonic waves propagated from left to 
right with velocity v. This is the expression for it 



indicating f = Acos-^ (y - vt) . 
L * 



110 LECTURE X. PART I. 

Molar. Hence, 

2-7T rf 2?r 

-- -- * = ---asm?; 



and the rate of doing work is 

47T* , 

aim sin q. 
X 

That is the rate at which this plane, working on the elastic solid 
on the right-hand side of it, does work (' per unit area of the 
plane" understood). Multiply this by dt and integrate through 
a period r = X >'. Now 



f sin'Vft = fl - 

Jo o * 



cos 2 ) ^ = dt = 



The rate of doing work then, per period, is 

27T 2 , 27rV>i 

x' a mr = x ' 

If it is possible for a set of waves to advance uniformly into 
space previously undisturbed, then it is certain that the work done 
per period must be equal to the energy in the medium per wave 
length. Let us then work out the energy per wave length. 

It is easily proved that, in waves in a homogeneous elastic 
solid, the energy is half potential of elastic stress, and half kinetic 
energy; and it will shorten the matter, simply to calculate the 
kinetic energy and double it, taking that as the energy in the 
medium per wave length. In our notation of yesterday, we took 

2 as the density. Multiply this by dy, to get the mass of an 

infinitesimal portion (per unit of area in the plane of the wave). 
The kinetic energy of this mass is 

IP , 1 y 2 2 . 



Integrating this through a wave length, and doubling it so as to 

get the whole energy, we have - L . Compare that with the 

A. 

work done per period, viz. - /, if J denote as vesterday the 
2. X 47r 

rigidity instead of n. We see that they are equal, because (velocity 



ENERGY OF WAVES. Ill 



of propagation) = A/ as we find from the elementary equation Molar. 
of the wave motion, 



Thii8 the work done por period is equal to the energy per wave 
length. 

This agrees with what we know from the ordinary general 
solution of the equation of motion by arbitrary functions that it 
is possible for a discontinuous series of waves to be propagated 
into the elastic medium, previously quiescent : and is coex- 
tensive with the case of velocity of propagation independent of 
wave length, for a regular simple harmonic endless succession of 
waves. But if our present energy equation did not verify, it 
would be impossible to have a discontinuous series of waves 
propagated forward without change of form into a medium 
previously quiescent. I wanted to verify the energy equation for 
the case of the homogeneous elastic solid, because we are con- 
cerned with a case in which this is not verified ; that is to say, 
when we put in our molecules In this case, the work done per 
period is less than the energy in the medium per wave length, 
and therefore it is impossible for the waves to advance without 
change of form. 

Before we go on to that, let us stay a little longer in a 
homogeneous elastic solid, and look at the well-known solution by 
discontinuous functions. The equation of motion is 



Although I said I would not formally prove this now, it is in 
reality proved by our old equation 



I took the liberty of asking Professor Ball two days ago whether 
he had a name for this symbol v* '> anu< ne nas mentioned to me 
null i. a humorous suggestion of Maxwell's. It is the name of an 
Egyptian harp, which was of that shape. I do not know that 
it is a bad name for it Laplacian I do not like for several reasons 
both historical and phonetic. [Jan. 22, 1802. Since 1884 I have 
found nothing better, and I now call it Laplacian.] 



112 LECTURE X. PART I. 

Molar. I should have told you that this is the case of a plane wave 

propagated in the direction of OF, with the plane of the wave 
parallel to XZ; for which case, nabla of (that is to say y 2 ) 

becomes simply , f . The time-honored solution of this equa- 
tion is 



where / and F are arbitrary functions. You can verify that by 
differentiation. This solution in arbitrary functions proves that a 
discontinuous series of waves is possible ; and knowing that a discon- 
tinuous series is possible, you could tell without working it out, that 
the work done per period by the medium on the one side of the 
plane which you take perpendicular to the line of propagation 
must be equal to the energy of the medium per wave length. 

Before passing on to the energy solution for the case in which 
we have attached molecules, in which this equality of energy and 
work does not hold, with the result that you cannot get the 
discontinuous single pulse or sequence of pulses, I want to suggest 
another elementary exercise for the anticipated arithmetical labo- 
ratory. It is to illustrate the propagation of waves in a medium 
in which the velocity is not independent of the wave length, and 
to contrast that with the propagation of waves when the velocity 
is independent of the wave length in order that you may feel 
for yourselves what these two or three symbols show us, but which 
we need to look at from a good many points of view before we 
can make it our own, and understand it thoroughly. To realize 

that this equation ~ = const, x ,- gives us constant velocities 
dr dy fe 

for all wave lengths, and that constant velocities for all wave 
lengths implies this equation, and to see that that goes along with 
the propagation of a discontinuous pulsation without change of 
figure, or a discontinuous succession of pulsations without change 
of character, I want an illustration of it, by the consideration of a 
case in which the condition of constancy of velocity for different 
wave lengths is not fulfilled. 

1 ask you first to notice the formula 

Hi-* 8 ) 



ENERGY OF WAVES. 



113 



which is familiar to all mathematical readers as leading up to Molar. 
Fourier's harmonic series of sines and cosines. It is proved by 
taking 

2 cos q = f + ~ l , 

and resolving S into two partial fractions. Poisson and others* 
make this series the foundation of a demonstration of Fourier's 
theorem. If e < 1 the series is convergent ; when e = 1 it ceases 

to converge. If we take q - and draw the curve whose de- 
pendent coordinate is x = S, what have we ? 

Take t = and measure off lengths from the origin 

tj = a, 2a, . . . 
The curve represented will be this (heavy curve). 




The heavy curve is 



5 3 cos 2-rry 
It is here drawn by the points 

(y,0-(0,l),(i A 

and symmetrical continuation. 
The dotted curve is 



,. 
5 4 cos 2-rry x 

It is here drawn by the points 

(y, *) = (o, f), (l -h\ (i, A), (i, i), 

and symmetrical continuation. 

I want the arithmetical laboratory to work this out and give 

* See Thomson and Tait'e Natural Philotophy, 77. 
T. L. 8 



114 



LECTURE X. PART 1. 



Molar. graphic representations of the periodic curves for several different 
values of e. The particular numerical case 
that I am going to suggest is one in which 
the curve will be more like this second 
curve which T draw ; it is much steeper 
and comes down more nearly to zero. Take 
the extreme case of e = I, and what happens ? 
S is infinitely great for q infinitely small, and 
is infinitely small for all other values of q less 
than a. For any value of e, the maximum 
and minimum ordinates of the curve (corre- 
sponding respectively to </ = 0, and q = 180) 
are 

1 +e 1 -e 



Molar: Di- 
gression, 
Deep-sea 
Waves. 



and therefore the minimum is 
(1 -.)/(! + *) 

of the maximum. Thus, if for example we take e = '9. we find the 
minimum ordinate to be 1 361 of the maximum. I suggest, as a 
mathematical exercise, to draw the curve for this case by the 
finite formula: and, as an arithmetical exercise, to calculate as 
man}- as you please of the ordinates by the series. You will find 
its convergence tediously slow. 

[(9) 43 -'0108, (-9)-":-: -0097]: 

you must take more than 43 or 44 terms to reach an accuracy of 
one per cent, in the result. So I do not think you will be inclined 
to calculate very many of the ordinates by the series. 

I would also advise those who have time to read Poisson's and 
Cauchy's great papers on deep-sea waves. (Poisson's Memoire sur 
la theorie des ondes. Paris, Mem. Acad. tici. I., LSI 6, pp. 71 18(i ; 
Annal. de Cliemie, v., 1817, pp. 122 142. Cuuchy, Memoire sur la 
theorie de la propagation des ondes a la surface d'un fluide pesant 
dune profondeur indefinie [1815], Paris, Mem. Sav. fitruwj. I., 
1827, pp. 3 312.) Those papers are exceedingly fine pieces of 
true mathematics; and they are very strong. But you might 
have the hydrodynamical beginnings presented much more fasci- 
natingly. If you know the elementary theory of deep-sea waves, 
well and good: then take Poisson and Cauchy for the higher 
analytical treatment. Those who do not know the theory of 



ENERGY OF WAVES. 115 

deep-sea waves may read it up in elementary books. The best Mola F : Di - 
text-books I know for Hydrokinetics are Besant's and Lamb's. Deep-sea 

The great struggle of 1815, not that fought out on the plains Wave8 - 
of Belgium, was, who was to rule the waves, Cauchy or Poisson. 
Their memoirs seem to me of very nearly equal merit. I have no 
doubt the judges had good reason for giving the award to Cauchy, 
but Poisson 's paper also is splendid. I can see that the two writers 
respected each other very much, and I suppose each thought the 
other's work as good as his own (?and sometimes better !). 

The problem which they solve is this, in their high analytical 
style: Every portion of an infinite area of water is started initially 
with an arbitrarily stated infinitesimal displacement from the 
level and an arbitrarily stated velocity up or down from the level, 
and the inquiry is, what will be the result ? It is obvious that 
you have the solution of that problem from the more elementary 
problem, what is the result of an infinitesimal displacement at 
a single point, such as may be produced by throwing a stone 
into water ? Let a solid, say, cause a depression in any place, the 
velocity of the solid performing the part of giving velocity and 
displacement to the surface of the water: then consider the solid 
suddenly annulled. The same thing in two dimensions is exceed- 
ingly simple Take, for example, waves in an infinitely deep 
canal with vertical sides. Take a sudden disturbance in the canal, 
equal over all its breadth, and inquire what will the result be? 

I wish now to help you toward an understanding of Cauchy's 
and Poisson 's solutions. They only give symbols and occasionally 
numerical results : they do not give any diagrams or graphical repre- 
sentations ; and I think it would repay any one who is inclined to go 
into the subject to work it out with graphic representations thus : 
first I must tell you that the elementary Hydrokinetic solution for 
deep-sea waves is simply a set of waves, or a set of standing vibra- 
tions (take which you please): the propagational velocity of the waves 



being * ' an( ^ therefore for different waves directly propor- 
tional to the square root of the wave-length ; and the vibrational 
period of waves or of standing vibrations being / , also 

directly proportional to the square root of the wave-length. 
Thus, so far as concerns only the varying shape of the disturbed 
water surface, the whole result of the elementary Hydrokinetics of 

82 



116 LECTURE X. PART I. 



Molar :Di- d e ep-sea wave-motion is expressed by one or other of these 



Deep-s 
Waves. 



sea equations which I write down ; 

2?r / _ / #X\ _ , sin ^TTJ/ sin 

Superposition of two motions represented by either formula 
gives a specimen of single motion represented by the other, as you 
all know well by your elementary trigonometry. 

And now, in respect to Poisson's and Cauchy's great mathe- 
matical work on deep-sea waves, it will be satisfactory to you 
to know that it consists merely in the additions of samples 
represented by whichever of these formulas you please to take. 
The simple formula, or summation of it with different values of h 
and X, represents waves with straight ridges, or generally straight 
lines of equal displacement : or, as we may call it : two-dimensional 
wave-motion. Every possible case of three-dimensional wave- 
motion (including the circular waves produced by throwing a 
stone into water) is represented by summation of the samples of 
the formula, as it stands, and with z substituted for y ; y and z 
representing Cartesian coordinates in the horizontal plane of the 
undisturbed water-surface, and x representing elevation of the 
disturbed water-surface above this plane. 

And now, confining ourselves to the two-dimensional wave- 
motion, I suggest to our arithmetical laboratory, to calculate and 
draw the curve represented by 

s = -| + e cos q l + e- cos 2</ 2 -f e 3 cos 3q 3 + ... e { cos iq,- + ... 

where <?, = (?/ V;t} ; with v f = - ... 

Calculate the curve corresponding to any values you please of 
e and of t. Also calculate and draw curves representing the sum 
of values of s with equal positive and negative values of t',-. 

I suggest particularly that you should perform this last 
described calculation for the cases, e=^ and e = ^. You have 
already the curves for t = in these cases shown on the diagram 
of page 113, and you will find it interesting to work out, in con- 
siderable detail, curves for other values of t which you can do 
without inordinately great labour, as the series are very rapidly 
convergent. You will thus have graphic representations of the 
two-dimensional case of Cauchy and Poisson's problem for infinitely 
deep water between two fixed parallel vertical planes. 



ENERGY OF WAVES. 117 

Lastly, I may say that if there are some among you who Molar: Di- 
will not shrink from the labour of calculating and adding forty or ^eep-sea 
fifty terms of the series, I advise you to do the same for e = '9 ; Waves, 
and you will have splendid graphical illustrations of the two- 
dimensioual problem of deep-sea waves initiated by a single 
disturbance along an endless straight line of water. If you do so, 
or if you spend a quarter-of-an-hour in planning to begin doing so, 
you will learn to thank Cauchy and Poisson for their magnificent 
mathematical treatment of their problem by definite integrals, 
and for their results from which, with very moderate labour, you 
may calculate the answer to any particular question that may be 
reasonably put with reference to the subject ; and may work out 
very thorough graphical illustrations of all varieties of the problem 
of deep-sea waves*. 

We are going to take our molecules again, and put them in the Molecular, 
ether; and look at the question, what is the velocity of propagation 
of waves through it under some suppositions which we shall make 
as to the masses of these embedded molecules, and how much they 
will modify the velocity of propagation from what it is in pure 
ether. Then we shall look at the matter, with respect to the 
question of the work done upon a plane perpendicular to the line 
of propagation, and we shall see that the energy per wave-length 
is greater than the work done per period, and that therefore it is 
impossible under these conditions for waves to advance uniformly 
into space previously occupied by quiescent matter. 

You will find, in Lord Rayleigh's book on sound, the question 
of the work done per period, and the energy per wave length, gone 
into: and the application of this principle, with respect to the 
possibility of independent suites of waves travelling without change 
of form, thoroughly explained. 

To-morrow we shall consider investigations respecting the 
difference of velocity of propagation in different directions in an 
aeolotropic elastic solid, for the foundation of the explanation of 
double refraction on mere elastic solid idea. The thing is quite 

* [Note of May, IH'.IM. For auuie of these see my papers: "On Stationary 
Waves iu Flowing Water," Phil. Mag. 1886, Vol. 22, pp. 353, 445, 517; 1887, 
Vol. 23, p. 52. "On the Front and Rear of a Free Procession of Waves in Deep 
Water," Phil. Mag. 1887, Vol. 23, p. 113; and, "On the Waves produced by a 
Single Impulse in Water of any depth or in a Dispersive Medium," Phil. Mag. 
1887, Vol. 23, p. 252.] 



118 LECTURE X. PART II. 



Molecular, familiar to many of you, no doubt, and you also know that it is a 
failure in regard to the explanation of the propagation of light in 
biaxal crystals. It is, however, an important piece of physical 
dynamics, and I shall touch upon it a little, and try to show it in 
as clear a light as I can. 



Ten minutes interval. 

Now for our proper molecular question. The distance from 
cavity to cavity in the ether is to be exceedingly small, in 
comparison with the wave-length, and the diameter of each 
cavity is to be exceedingly small, in comparison with the distance 
from cavity to cavity. Let the lining of the cavity be an 
ideal rigid massless shell. Let the next shell within be a rigid 

shell of mass -- 1 a . I represent the thing in this diagram as 

if we had just two of these massive shells 
and a solid nucleus. The enormous mass of 
the matter of the grosser kind which exists 
in the luminiferous ether when permeated 
by even such a comparatively non-dense 
body as air, would bring us at once to very 
great numbers in respect to the masses which 
we will suppose inside this cavity, in comparison with the masses 
of comparable bulks of the luminiferous ether, if there is time 
to-morrow, we shall look a little to the possible suppositions as to 
the density of the luminiferous ether, and what limits of greatness 
or smalluess are conceivable in respect to it. At present we have 
enough to go upon to let us see that, even in air of ordinary 
density, the mass of air per cubic centimetre must be enormously 
great, in comparison with the mass of the luminiferous ether per 
cubic centimetre. We must therefore have something enormously 
massive in the interior of these cavities. We shall think a good 
deal of this yet, and try to find how it is we can have the large 
quantity of energy that is necessary to account for the heating of 
a body such as water by the passage of light through it, or for 
the phosphorescence of a body which is luminous for several days 
after it has been excited by light. I do not think we shall have 
the slightest difficulty in explaining these things. These are not 




DIFFICULTIES IN WAVE THEORY OF LIGHT. 119 

the difficulties. The difficulties of the wave theory of light are Molecular. 
difficulties which do not strike the popular imagination at all. 
They are the difficulties of accounting for polarization by reflection 
with the right amount of light reflected ; and of accounting for 
double refraction with the form of wave-surface guessed by 
Huyghens and proved experimentally by Stokes. With the 
general character of the phenomena we have no difficulty what- 
ever ; the great difficulty, in respect to the wave theory of light, 
is to bring out the proper quantities in the dynamical calculation 
of these effects. 

There is no difficulty in explaining the energy required for 
heating a body by radiant heat pawing through it, nor how it is 
that it sometimes comes out as visible light and, it may be, so 
slowly that it may continue appearing as light for two or three 
days. All these properties, wonderful as they are, seem to come 
as a matter of course from the dynamical consideration. So much 
so that any one not knowing these phenomena would have dis- 
covered them on working out the subject dynamically. He would 
discover anomalous dispersion, fluorescence, phosphorescence, and 
the well-known visible and invisible radiant heat of longer periods 
emitted by a body which has been heated and left to cool. All 
these phenomena might have been discovered by dynamics ; and 
a dynamical theory that discovers what is afterwards verified by 
experiment is a very estimable piece of physical dynamics. 

I speak with confidence in this subject because I am ashamed 
to say that 1 never heard of anomalous dispersion until after I 
found it lurking in the formulas. And, when I looked into the 
matter, 1 found to my shame that a thing which had been known 
by others for tifteen or twenty years* I had not known until I 
found it in the dynamics. 

Take our concluding formula of yesterday (p. 100 above), with 
some changes of notation 



where denotes the propagational velocity of waves of period r ; 

Leroux, " Dispersion anomale de la vapeur d'iode," Comptet llendut, LV., 1862, 
pp. 1'26 128: ; ',.</.;. Ann. civil., 1802, pp. 659, 660. Christiansen, " Ueber die 
Brechungtiverhaltnisse einer wehiKciKtigen Losung des FuchninB," Ann. I'hyt. Chem. 
CXLI, 1870, pp. 479, 480: I'hil. May., xu., 1871, p. 244; Annulet de Chimir, xxv., 
1872, pp. 213, 214. 



120 LECTURE X. PART II. 

Molecular, n, p, and m l denote now respectively the rigidity of the ether, 
the mass of the ether in unit volume of space, and the sum 
of the masses of the first interior shells of the embedded 
molecules in unit volume of space ; 

d the force of the first spring, per unit elongation, multiplied by 
the number of molecules embedded in the ether per unit 
volume of space ; 

K, K , K , &c., in order of ascending magnitude, the fundamental 
periods of the molecule when the outer shell is held fixed ; 

R, R,, R /t , &c., denote for the separate fundamental vibrations the 
ratio of the energy of the first interior shell to the whole 
energy of the complex vibrator. 

Let us consider what the wave-period r may be relatively to 
the fundamental periods K, K t , *,... of the vibrator on the sup- 
position of the bounding shell held fixed, to give us a good 
reasonable explanation of dispersion, in accordance with the facts 
of observation with respect to the difference of velocity for 
different periods. To help us with this consideration, take our 
previous auxiliary formulas 

1 _ p GIT- x l \ 

= n " 



where and x 1 denote respectively the simultaneous maximum 
displacements of the outermost massless shell and the first of the 
massive shells within it. 

If r were less than the smallest of the fundamental periods, 

TT would be negative, the wave- velocity would be greater than in 

free ether, and the refractive index would be less than unity. 
But in all known cases the refractive index is greater than unity ; 

and when this is so, -j - 1 must be positive. 1 want to see if 

we can get our formula to cover a range, including all light from 
the highest ultra-violet photographic light of about half the 
wave-length of sodium light down to ^the lowest we know of, 
which is the radiant heat from a Leslie cube with a wave-length 



KEFRACTIVITY THROUGH WIDE RANGE. 121 

that I hear from Prof. Langley since I spoke to you on the Molecular 
subject a week ago is about y^jy of a centimetre or 17 times the 
wave-length of sodium light. That will be a range of about 
forty to one. The highest invisible ultra-violet light hitherto 
determined, by its photographic action, has a period about 1/40 
of the period of the lowest invisible radiation of radiant heat that 
has yet been experimented upon. 

It is probable that all or many colourless transparent liquids 
and solids are mediums for which throughout every part of that 
range there are no anomalous dispersions. I think it is almost 
certain that for rock-salt, in the lower part of the range, there 
are no anomalous dispersions at all. In fact Langley 's experiments 
on radiant heat are made with rock-salt ; and in all experiments 
made with rock-salt, it seems as if little or no radiant heat is 
absorbed by it. At all events, we could not be satisfied unless we 
can show that this kind of supposition will account for dispersion 
through a range of period from one to forty. It is obvious that 
if we are to have continuous refraction without anomalous dis- 
persion through that wide range of periods, there cannot be any 
of the periods K, K t , *, ... within it. 

To-morrow we shall consider the case in which the wave-period 
is longer than the longest of the molecular periods; and we shall 
find that on this supposition our formula serves well to represent 
all we have hitherto known by experiment regarding ordinary 
dispersion. [Added July 7, 1898. This was true in October 1884 ; 
but measurements by Langley of the refractivity of rock-salt for 
radiant heat of wave-lengths (in air or ether) from '43 of a mikron 
to 5'3 mikrons, (the " mikron " being 10~* of a metre or 10~ 4 of a 
centimetre), published in 1886 (Phil. Mag. 1886, 2nd half-year), 
showed that there must be a molecular period longer than that 
corresponding to wave-length 5'3. In an addition to Lecture 
XII, Part II, we shall see that subsequent measurements of 
refractivity by Rubens, Paschen, and others, extend the range 
of ordinary dispersion by rock-salt to wave-lengths of 23 mikrons; 
and give results in splendid agreement with our formula, which 
is identical with Sellraeier's expression of his own original theory, 
through a range of from '4 of a mikron to 23 mikrons, and 
indicate 56 mikrous as being the probable wave-length of radiant 
heat in ether of which the period is a critical period for rock-salt.] 



LECTUKE XL 

FRIDAY, October 10, 3.30 P.M. 

WE shall now take up the subject of an elastic solid which is 
not isotropic. As 1 said yesterday, we do not find the mere 
consideration of elastic solid satisfactory or successful for explain- 
ing the properties of crystals with reference to light. It is, how- 
ever, to my mind quite essential that we should understand all 
that is to be known about homogeneous elastic solids and waves 
in them, in order that we may contrast waves of light in a crystal 
with waves in a homogeneous elastic .solid. 

Aeolotropy is in analogy with Cauchy's word isotropy which 
means equal properties in all directions. The formation of a word 
to represent that which is not isotropic was a question of some 
interest to those who had to speak of these subjects. I see the 
Germans have adopted the term anisotropy. If we used this in 
English we should have to say : " An anisotropic solid is not an 
isotropic solid " ; and this jangle between the prefix an (privative) 
and the article an, if nothing else, would prevent us from adopting 
that method of distinguishing a non-isotropic solid from an 
isotropic solid. 1 consulted my Glasgow University colleague 
Prof. Lushington and we had a good deal of talk over the subject. 
He gave me several charming Greek illustrations and wound up 
with the word aeolotropy. He pointed out that <u'o\ov moans 
variegated; and that the Greeks used the same word for variegated 
in respect to shape, colour and motion ; example of this last, our 
old friend " KopvOaioXo? "ETo>p." There is no doubt of the 
classical propriety of the word and it has turned out very con- 
venient in science. That which is different in different directions, 
or is variegated according to direction, is called aeolotropic. 

The consequences of aeolotropy upon the motion of waves, or 
the equilibrium of particles, in an elastic solid is an exceedingly 



AEOLOTROPY IN AN ELASTIC SOLID. 123 

interesting fundamental subject in physical science ; so that there Molar, 
is no need for apology in bringing our thoughts to it here except, 
perhaps, that it is too well known. On that account 1 shall be 
very brief and merely call attention to two or three fundamental 
points. I am going to take up presently, as a branch of molar 
dynamics, the actual propagation of a wave ; and in the mathe- 
matical investigation, I intend to give you nothing but what is 
true of the propagation of a plane wave in an elastic solid, not 
limited to any particular condition of aeolotropy ; in an elastic solid, 
that is to say, which has aeolotropy of the most general kind. 

Before doing that, which is strictly a problem of continuous or 
molar dynamics, I want to touch upon the somewhat cloud-land 
molecular beginning of the subject, and refer you back to the old 
papers of Navier and Poisson, in which the laws of equilibrium or 
motion of an elastic solid were worked out from the consideration 
of points mutually influencing one another with forces which are 
functions of the distance. There can be no doubt of the mathe- 
matical validity of investigations of that kind and of their interest 
in connection with molecular views of matter; but we have long 
passed away from the stage in which Father Boscovich is accepted 
as being the originator of a correct representation of the ultimate 
nature of matter and force. Still, there is a never-ending interest 
in the definite mathematical problem of the equilibrium and motion 
of a set of points endowed with inertia and mutually acting upon 
one another with any given forces. We cannot but be conscious 
of the one splendid application of that problem to what used to be 
called physical astronomy but which is now more properly willed 
dynamical astronomy, or the motions of the heavenly bodies. But 
it is not of these grand motions of mutually attracting particles 
that we must now think. It is equilibriums and infinitesimal 
motions which form the subject of the special molecular dynamics 
now before us. 

Many writers [Navier (1827), Poisson (1828), Cauchy, F. Neu- 
mann, Saint- Venant, and others] who have worked upon this 
subject have come upon a certain definite relation or set of 
relations between moduluses of elasticity which seemed to them 
essential to the hypothesis that matter consists of particles acting 
upon one another with mutual forces, and that the elasticity of a 
solid is the manifestation of the forcive required to hold the 
particles displaced infiiiitesiinally from the position in which the 



124 LECTURE XI. 

mutual forces will balance. This, which is sometimes called 
Navier's relation, sometimes Poisson's relation, and in connection 
with which we have the well-known "Poisson's ratio," I want to 
show you is not an essential of the hypothesis in question. Their 
supposed result for the case of an isotropic body is interesting, 
though now thoroughly disproved theoretically and experimentally. 
Doubtless most of you know it ; it is in Thomson and Tait, and 
I suppose in every elementary book upon the subject. I will just 
repeat it. 

An isotropic solid, according to Navier's or Poisson's theory, 
would fulfil the following condition : if a column of it were pulled 
lengthwise, the lateral dimensions would be shortened by a 
quarter of the proportion that is added to the length ; and the 
proportionate reduction of the cross-sectional area would therefore 
be half the proportion of the elongation. Stokes called attention 
to the viciousness of this conclusion as a practical matter in 
respect to the realities of elastic solids. He pointed out that jelly 
and india-rubber and the like, instead of exhibiting lateral 
shrinkage only to the extent of one quarter of the elongation, give 
really enough of shrinkage to cause no reduction in volume at all. 
That is to say, india-rubber and such bodies vary the area of the 
cross-section in inverse proportion to the elongation so that the 
product of the length into the area of the cross-section remains 
constant. Thus the proportionate linear contraction across the 
line of pull is half the elongation instead of only quarter as 
according to Navier and Poisson. 

Stokes* also referred to a promise that 1 made, 1 think it 
was in the year 1856, to the effect that out of matter fulfilling 
Poisson's condition a model may be made of an elastic solid, 
which, when the scale of parts is sufficiently reduced, will be a 
homogeneous elastic solid not fulfilling Poisson's condition. That 
promise of mine which was made 30 years ago, I propose this 
moment to fulfil, never having done so before. 

Let this box help us to think of 8 atoms placed at its 8 
corners, with the box annulled. The kind of elastic model I am 
going to suppose is this : particles or atoms arranged equidistantly 
in equidistant parallel rows and connected by springs in a certain 

* [lleport to Brit. Association, Cambridge, 1C'2. " Ou Double llefiaction, " 
p. '202, at bottom.] 



OREF\'S THEORY REALISED TV MECHANICAL MODEL. 125 

definite way. I am going to show you that we can connect Molar. 
neighbouring particles of a Boscovich elastic solid with a special 
appliance of cord, and a sufficient number of springs, to fulfil the 
condition of giving 18 independent moduluses ; then by trans- 
forming the coordinates to an orientation in the solid taken at 
random, we get the celebrated 21 coefficients, or moduluses, of 
Green's theory. I suppose you all know that Green took a short- 
cut to the truth ; he did not go into the physics of the thing at 
all, but simply took the general quadratic expression for energy in 
terms of the 6 strain components, with its 21 independent coeffi- 
cients, as the most general supposition that can be made with 
regard to an elastic solid. 

To make a model of a solid having the 21 independent coeffi- 
cients of Green's theory, think of how many disposable springs we 
have with which to connect 8 particles at the corners of a parallel- 
epiped. Let them be connected by springs first along the 12 
edges of the parallelepiped. That clearly will not be sufficient to 
give any rigidity of figure whatever, so far as distortions in the 
principal planes are concerned. These 12 springs connecting in 
this way the 8 particles would give resistances to elongations 
in the directions of the edges ; but no resistance whatever to 
obliquity; you could change the configuration from rectangular, 
if given so as in the box before you, into an oblique parallelepiped, 
and alter the obliquity indefinitely, bringing if you please all 
the 8 atoms into one plane or into one line, without calling any 
resisting forces into play. What then must we have, in order 
to give resistance against obliquity ? \Ve can connect particles 
diagonally. We have in the first place, the two diagonals in each 
face although we shall see that the two will virtually count as but 
one ; and then we have the four body diagonals. 

Now let me see how many disposables we have got. Remark 
that each edge is common to four parallelepipeds of the Boscovich 
assemblage. Hence we have only a quarter of the number of 
the twelve edge springs independently available. Thus we have 
virtually three disposables from the edge springs. Rich face is 
common to two parallelepipeds ; therefore from the two diagonals 
in each face we have only one disposable, making in the six faces, 
six disposables. We have the four body diagonals not common to 
any other parallelepipeds and therefore four disposables from them. 
We have thus now 13 disposables in the stiffnesses of 13 springs. 



126 LECTURE XI. 

And we have two more disposables in the ratios of edges of 
the parallelepiped. Lastly we have three angles of three of the 
oblique parallelograms constituting the faces of our oblique parallel- 
epiped. Thus we have in all 18 disposables. But these 18 
disposables cannot give us 18 independent moduluses because it is 
obvious that they cannot give us infinite resistance to compression 
with finite isotropic rigidity, a case which is essentially included 
in 18 independent moduluses. Hence I must now find some 
other disposable or disposables that will enable me to give any 
compressibility I please in the case of an isotropic solid, and to 
give Green's 21 independent coefficients, for an aeolotropic solid. 
For this purpose we must add something to our mechanism that 
can make the assemblage incompressible or give it any compres- 
sibility we please ; so that, for example, we can make it represent 
either cork or india-rubber, the extremes in respect to elasticity of 
known natural solids. 

I must confess that since 1856 when I promised this result I 
have never seen any simple definite way of realising it until a few 
months ago when in making preparations for these lectures I 
found I could do it by running a cord twice round the edges of 
our parallelepiped of atoms as you see me now doing on the model 
before you. It is easier to do this thus with an actual cord and 
with rings fixed at the eight corners of a cubic box, than to 
imagine it done. There is a vast number of ways of doing it ; 
I cannot tell you how many, I wish I could. It is a not unin- 
teresting labyrinthine puzzle to find them all and to systematise 
the finding. You see now we are finding one way. 

Here it is expressed in terms of the coordinates of the corners 
as we have taken them in succession. 

(000) (001) (Oil) (010) (000) (001) (011)(010)(000)(100)(110)(010) 



[April 14, 1898. For the accompanying very clear diagram 
(fig. 1) representing another of the vast number of ways of laying 
a cord round the edges of a parallelepiped, I am indebted to 
M. Brillouin who has added abstracts of some of the present 
lectures to a translation by M. Lugol of Vol. I. of my Popular 
Lectures and Addresses. M. Brillouin describes his diagram as 
follows : " J'ai modifie 1'ordre indique par Thomson en permutant 
le 6 e et le 8 e sommet, pour que la corde suive chaque cote en sens 



GREENS THEORY REALISED IN MECHANICAL MODEL. 



127 



oppose* & ses deux trajets. Dana 1'ordre primitif, la face (000) Molar. 
(001) (Oil) (010) e*tait parcourue deux fois dans le ineuie sena"] 

You see I have drawn the cord just three times through each 
corner ring and I now tighten it over its whole length and tie the 
ends together. Remark now that if the cord is inextensible it 
secures that the sum of the lengths of the 12 edges of the parallel- 
epiped remains constant, whatever change be given to the relative 
positions of the 8 corners. This condition would be fulfilled for 
any change whatever of the configuration ; but it is understood 
that it always remains a parallelepiped, because our application of 
the arrangement is to a homogeneous strain of an elastic solid 
according to Boscovich. A cord must similarly be carried twice 
round the 4 edges of every one of the contiguous parallelepipeds ; 
and eight of these have a common corner, at which we suppose 
placed a single ring. Hence every ring is traversed three times 



' 




(100) 



(010) 



(110) 



by each one of eight endless cords. Each edge is common to four 
contiguous parallelepipeds and therefore it has two portions of 
each of four endless cords j>assing along it. 

Suppose now for example the parallelepiped to be a cube. 
Ine.xU-nsible cords applied in the manner described between 
neighbouring atoms, keep constant the sum of the lengths of the 
12 edges of each cube, and therefore secure that its volume is 
constant for every infinitesimal displacement. Consider for a 
moment the assemblage of ring atoms thus connected by endless 
cords. In itself and without further application to molecular 
theory we see a very interesting stnicture which, provided the 
cords are kept stretchrd, occupies a constant volume of space and 
yet is perfectly without rigidity for any kind of distortion. You 



128 



LECTURE XT. 



see if I elongate it in one of the three directions of the parallel 
edges of the cubes, it necessarily shrinks in the perpendicular 
directions so as to keep constant volume. This kind of deforma- 
tion gives us two of the five components of distortion without 
change of bulk ; and the other three are given by the shearings 
which we have called a, b, c ; of which, for instance, a is a distor- 
tion such that the two square parallel faces of the cube perpen- 
dicular to OX become rhombuses while the other four remain 
square. 

[April 14, 1898. The cube (fig. 1) with an endless cord twice 
along each edge is, at least mechanically, somewhat interesting in 
realising an isolated mechanism for securing constancy of volume 
of a hexahedron, without other restriction of complete liberty of 
its eight corners than constancy of volume requires: provided only 
that the hexahedron is infinitely nearly cubic, and that each face 
is the (plane or curved) surface of minimum area bounded by four 
straight lines. Two years ago in preparing this lecture for the 
press from Mr Hathaway 's papyrograph, a much simpler mechan- 
ism of cords for securing constancy of volume, occupied by an 
assemblage of a vast number of points given in cubic order, than 
is provided by the linking together of cubes separately fulfilling 
this condition, occurred to me. It was not till some time later 




Fig. 2. 

that I found myself anticipated by M. Brillouin. I am much 
pleased to find that he has interested himself in the subject and 
has introduced a new idea, by which the condition of constant 
volume is realised in the exceedingly simple and beautiful mechan- 
ism of endless cords represented in the annexed diagram (fig. 2), 
copied from one of his articles in the volume already referred to. 
This diagram represents endless cords of which just one is shown 



GREEN'S THEORY REALISED IN MECHANICAL MODEL. 129 



complete. These cords pass through rings not shown in fig. 2. Molar. 
Fig. 3 shows one of the rings viewed in the direction of a diagonal 
of the cube and seen to be traversed twelve times by four endless 
cords of which one is shown complete. In an assemblage of a vast 
number of rings thus connected by endless cords and arranged in 
cubic, or approximately cubic, order, one set of three conterminous 
edges of each cube is kept constant, and therefore for the exactly 




Fig. 3. 

cubic order the volume is a maximum. This method, inasmuch 
as it implies only two lines of cord along each edge common to 
four cubes, is vastly simpler than the original method described 
above as in my lecture at Baltimore, which required eight lines of 
cord along each common edge. The kinematic result with its 
dynamic consequences, is the same in the two methods. 

To avoid the assumption of an inextensible elastic cord, place 
at each corner, common to eight cubes, six bell-cranks properly 
pivoted to produce the effect of cords running, as it were, round 
pulleys, which we first realised by the six cords running through 
the ring of fig. 3. And instead of each straight portion of cord, 
substitute an inextensible bar of rigid matter, hooked at its two 
ends to the arm-ends of the proper bell-cranks ; just as are the 
two ends of each copper bell-wire, so well known in the nineteenth 
and preceding centuries, but perhaps to be forgotten early in the 
T.L. 9 



130 LECTURE XI. 

twentieth century, when children grow up who have never seen 
bell-cranks and bell-wires and only know the electric bell. Remark 
now that our connecting-rods, being rigid, can transmit push as 
well as pull ; instead of merely the pull of the flexible cord 
with which we commenced. Our model now may be constructed 
wholly of matter fulfilling the Poisson-Navier condition, and it 
gives us a molecular structure for matter violating that condition. 

Suppose now we ideally introduce repulsions in the lines of 
the body diagonals between the four pairs of corners of each cube. 
This will keep stretched, all the cords between rings, or con- 
necting-rods between arm-ends of bell-cranks ; and will give a 
cubically isotropic elastic solid with a certain definite aeolotropic 
quality. If besides we introduce mutual forces in the edges of 
the cubes between each atom and its nearest neighbour, we can 
give complete isotropy, or any prescribed aeolotropy consistent 
with cubic isotropy. All this is for an incompressible solid. But 
lastly, by substituting india-rubber elastics for the cords, or ideal 
attractions or repulsions (Boscovichian) instead of the connecting- 
rods between bell-crank-arm-ends, we allow for any degree of 
compressibility, and produce if we please a completely isotropic 
elastic solid with any prescribed values for the moduluses of 
rigidity and resistance to compression, fulfilling or not fulfilling 
Poisson's ratio. It is mechanically interesting to work out details 
for this problem for the case suggested by cork, that is, a perfectly 
isotropic elastic solid, having rigidity much greater in proportion 
to resistance to compression than in the case of Poisson's ratio, 
and just sufficient to produce constancy of cross-section in a 
column compressed, or elongated, by forces applied to its ends. 
It is also interesting to go further and produce a solid of which a 
column shall shrink transversely when compressed merely by 
longitudinal force. 

In my lecture at Baltimore I indicated, without going into 
details, how by taking a parallelepiped of unequal edges instead 
of a cube and introducing different degrees of elasticity in the 
portions of the cords lying along the different edges of each 
parallelepiped, and by introducing also forces of attraction or 
repulsion among neighbouring atoms, we can produce a model 
elastic solid with the twenty-one independent moduluses of 
Green's theory. 

But the method by cords and pulleys or bell-cranks which we 



GREEN'S THEORY REALISED IN MECHANICAL MODEL. 131 

have been considering, though highly interesting in mechanics, Molar, 
and dignified by its relationship to Lagrange's original method of 
proving his theorem of " virtual velocities " (law of work done) in 
mathematical dynamics, lost much of its interest for molecular 
physics when I found* that the restriction to Poisson's ratio in 
an elastic solid held only for the case of a homogeneous assemblage 
of single Boscovich point-atoms: and that, in a homogeneous as- 
semblage of pairs of dissimilar atoms, laws of force between the 
similar and between the dissimilar atoms can readily be assigned, 
so as to give any prescribed rigidity and any prescribed modulus of 
resistance to compression for an isotropic elastic solid : and for 
an aeolotropic homogeneous solid, Green's 21 independent modu- 
luses of elasticity, or 18 when axes of coordinates are so chosen 
as to reduce the number by three.] 

I want now to go through a piece of mathematical work with 
you which, though indicated by Green^, has not hitherto, so far 
as I know, been given anywhere, except partially in my Article 
on "Elasticity" in the Encycloptedia Britannica. It is to find the 
most general possible plane wave in a homogeneous elastic solid of 
the most general aeolotropy possible, expressed in terms of Green's 
21 independent modulusesj. Taking Green's general formula 

[Proc. R. S. K. July 1 and 15, 1889 : Math, and Phyi. Papert, Vol. m. Art. 
xcni., p. 395 : also " On the Elasticity of a Crystal according to Boscovich," Proc. 
R. S., Jane 15, 1893, and republished as an Appendix to present volume.] 

t Green's Mathematical Papert (Macmillan, 1871), pp. 307, 308. 

J [June 16, 1898. Through references in Todhnnter and Pearson's Elasticity 
I have recently found three very important and suggestive memoirs by Blanchet in 
Liourillf'* Journal, Voln. v. and vn. (1840 and 1842), in which this problem is 
treated on the foundation of 36 independent coefficients in the six linear equations 
expressing each of the six stress-components in terms of the six strain-components. 
In respect to the history of the doctrine of energy in abstract dynamics, it is curious 
to find in a Report to the French Academy of Sciences by Poisson, Coriolis, and 
Sturm (Comptet Rtndiu, Vol. vn. , p. 1143) on the first of these memoirs (which had 
been presented to the Academy on August 8, 1838), the following sentence : " Les 
equations differentielles auxquelles sont assuje'tis les deplacements d'nn point 
quelconque du milieu ecarte de sa position d'equilibre renferment 36 coefficients 
constants, qni dependent de la nature du milieu, et qu'on ne pourrait rfduire a un 
moindre nombre tant faire det hypothettt tur la ditporition det nioKculet et tur let 
loit de leurt actiont mutuellei" (The italicising is mine.) Blanchet's second 
memoir, also involving essentially 36 independent coefficients, was presented to the 
Academy of Sciences on June 14, 1841, and was reported on by Cauchy, Liouville, 
and Duhamel without any protest against the 36 coefficients. In Green's memoir 
"On the Propagation of Light in crystallized Media," read May 20, 1839 to the 
Cambridge Philosophical Society, the expression for the energy of a strain as a 

92 



(1). 



132 LECTUBE XI. 

with the notation with which I put it before you in Lecture II. 

Part I. (pp. 22, 23, 24), we find 

P = lie + 12/+ 130 + 14a + 156 + 16c' 
Q = I2e + 22/+ 230 + 24a + 256 + 26c 
R = 13e + 23/+ 330 + 34a + 356 + 36c 
$ = 14e + 24/ + 340 + 44a + 456 + 46c 
T = I5e + 25/+ 350 + 45a + 556 + 56c 
U = IQe + 26/+ 360 + 46a + 566 + 66c 
Now, considering an ideal infinitesimal parallelepiped of the 

solid BacSy&z, remark that, in virtue of the stress components 

P, Q, E, S, T, U, it experiences pairs of opposing forces parallel to 

OX on its three pairs of faces as follows, 



\ dz 



and the total resultant component parallel to OX is therefore 
fdP dU dT^ 



Hence if we denote by (x -f , ?/ + rj, z + ) the coordinates at 
time t of a point of the solid of which (x, y, z} is the equilibrium, 
we have, as we found in Lecture II. (p. 26 above), 

d*% _ dP dU 
p dt 2 ~ dx + ~dy + ~dz ' 



r 




dx dy 
d?_dT dS 
P dt* ~ dx + dy 

Now a plane wave, or a succession of plane waves, or the 
motion resulting from the superposition of sets of plane waves 

quadratic function of the six strain-components had been fundamentally used ; and 
by it the fifteen equalities among the 36 coefficients in the linear equations for stress 
in terms of strain, reducing the number of independent coefficients to 21, had been 
demonstrated without hypothesis.] 



GREEN'S THEORY REALISED IN MECHANICAL MODEL. 133 



travelling in the same or in contrary directions, may be defined Molar. 
generally as any motion of the solid in which every infinitely thin 
lamina parallel to some fixed plane experiences a motion which is 
purely translational ; or in other words, a motion in which f, ij, f 
are functions of (p, t), where p denotes the perpendicular from 
to the plane through (x, y, z) parallel to the wave-front. If I, m, n 
denote the direction-cosines of this perpendicular, we have 



p = Ix + my -f nz 



(4), 



and therefore 



d d d _ d d d 

""5"~ v ~"j"~ \ ~j~~ TH ~| 1 ~p~ == ft ~ 

dx dp dy dp dz dp 
when these symbols are applied to any function of (p, t). 

Hence, according to the definitions of e, f, g, a, b, c which I 
gave you in Lecture II. (pp. 22, 23), we have 



-r, 
dp' 



<J=n~i 
y dp 



a = n -. + m 
dp 



-p -I- n ~ ; c 
dt) av 



'p ap <[p dp, 

Hence by (3), (5), (1) and (6) we find 



^1 

'//< 



- (6). 



dP dp' 



(7), 



where 



A =!!/' + u6/u j -I- 55w + 2 x 16/m + 2 x 56mn -f 2 x 
B = 66/ 1 -I- 22wi + 44-n 8 + 2 x 26/wi + 2 x 24mn f 2 x 4Gnl 
C = 55/ f 44m a -I- 33w -f- 2 x 4o/m -I- 2 x 34;/i + 2 x 35n/ 
4' = 56/ 1 + 24m 8 -I- 34u s + (25 + 46) Im -I- (23 + 44)mn +(36-f-45)n/ 
R = 15J' + 4Gm + 35n + (14 -f 56) Im -I- (36 + 45)mn +(13+ 55) nf 
C" = 16^ + 26m s -f- 45n -|- (12+ 66) Im + (25 + 46) inn + (14 + 56) nl, 

(8). 



Now, for our plane waves travelling in either direction, f, 17, 



134 LECTURE XL 

Molar. must each be a function of (p vt), where v denotes the propaga- 
tional velocity of the wave : hence 

Hence if we denote the acceleration components by jf, i), , 
equations (7) become 

C'ri + B''t, =0| 

A'l =0 (10). 



The determinant of these equations equated to zero gives, for 
v 2 , three essentially real values vf, v s 2 , v s 2 ; which are essentially 
positive if the coefficients 11, 66, &c., are of any values capable of 
representing the elastic properties of a stable elastic solid. And 
for each value of v 2 , equations (10) give determinate values, X, p, 
for the ratios T//^, / which we may denote by \, , //, ; X 2 , /*.,; 
\a, fi 3 \ so that finally we have, for the complete solution of our 
problem, superimposed sets of three waves expressed as follows, 

= fr (p + t\t) + </>, (p + v,t) + <f> 3 (p + vj) 



17 = X, (^ + ^,) + X, (4>, 



f l) + to (</> 2 + ^') + /*3 (03 + 

where 0J, 0.,, 0.,, -^j, -^r.,, -^r., denote arbitrary functions. 

This solution and the relative formulas will be very useful to 
us to-morrow when we shall be considering the corresponding 
" wave-surface " in all its generality ; that is to say, the surface 
touched by planes perpendicular to (I, >n, n) and at distances from 
an ideal origin of disturbance equal to vj, v z t, v 3 t. 



LECTURE XII. 
FRIDAY, October 10, 5 P.M. 

WE will look a little more at this wave problem. Our conclusion Molar. 
is, that if you choose arbitrarily, in any position whatever relatively 
to the elastic solid, a set of parallel planes for wave-fronts, there 
are three directions at right angles to one another (each generally 
oblique to the set of planes) which fulfil the condition, that the 
elastic force is in the direction of the displacement ; and the 
equations we have put down express the wave-motion. Each of the 
three waves will be a wave in which the oscillation of the matter 
in its front is as I am performing it now, i.e., an oscillation to and 
fro in a line oblique to the plane of the wave-front, represented by 
this piece of cardboard which I hold in my hand. You will find 
the vibrations of the three waves corresponding to the three roots 
of the determinantal cubic, whether they are oblique or not oblique 
to the wave-front, are in directions at right angles to one another. 

[Thirteen and a half years interval. Here is a very short 
proof. In equations (10) of Lecture XI., put 



and 



With this notation equations (10) give 
.sV 



" 

Multiplying these by Ja, Jft, -Jy respectively and adding; and 
dividing both sides of the resulting equation by S, we find 



136 LECTURE XII. PART I. 

This is a form of the deterrainantal cubic for the reduction of 
a homogeneous quadratic function of three variables, which I gave 
fifty-three years ago in the Cambridge Mathematical Journal 
(Math, and Physical Papers, Art. xv. Vol. I. p. 55). 

Writing down this equation for roots i^ 2 , v 2 -, and taking the 
difference, we find 



= P ^ - 



.(15). 



Hence if vf, v.? are equal, the second factor of this expression may 
have any value. If they are unequal it must be zero, and (13) 
gives 

ftfe + ^ + fcfc-O (16), 

which shows that the lines of the vibration in any two of our 
three waves are necessarily perpendicular to one another*, except 
in the case when the two propagational velocities are equal. In 
this case the two waves become one, and the line of vibration may 
be in any direction in a plane perpendicular to the line given by 
the third root. The case of three equal roots may also occur : in 
it the three waves become one, and the line of vibration may be 
in any direction whatever. Both in this case and in the case of 
two equal roots, each particle may describe a circular or elliptic 
orbit, or may move to and fro in a straight line. One equation 
among (I, m, n) gives a cone, such that, for the plane wave-front 
perpendicular to any one of its generating lines, two of the three 
wave- velocities are equal. Two equations among (I, m, u) give a 
line normal to a wave-front, or wave-fronts, for which the three 
wave-velocities are equal.] 

The consideration of the three sets of plane waves with three 
different propagational velocities, but with their fronts all parallel 
to one plane, leads us to a wave-surface different, so far as I know, 

This important proposition does not hold for the three directions of vibration 
found by Blanchet (footnote above), which, for three unequal roots of his cubic, 
are necessarily not all at right angles to one another unless Green's fifteen 
equalities are all fulfilled. Compare Thomson and Tail's Natural Philosophy, 
344, 345. 



BLA.NCHET'S THREE-SHEET WAVE. 137 

from anything that has been worked out hitherto in the dynamics Molar. 
of elastic solids a wave-surface in which there will be three 
sheets instead of only two, as in Fresnel's wave-surface : and in 
which there will be condensation and rarefaction at each point of 
each sheet, instead of the pure distortion of the ether at every 
point of each of the two sheets of Fresnel's wave-surface. It is 
a geometrical problem of no contemptible character to work out 
this wave-surface. 

[May 16, 1898. Here is the problem fully worked out, except 
the performance of the final elimination of I, m, n. In (14) 
above, for v put Ix + my + m and for n, wherever it occurs, put 
JlP m*. Let $ (I, m, x, y, z) denote what the second member 
of (14) then becomes. Take the following three equations : 



and eliminate I, m between them. The resulting equation ex- 
presses the wave-surface; that is to say, the surface, whose tangent 
plane at points of it where the direction cosines are I, in, n, is at 
distance v from the origin. I need scarcely say that a symmetrical 
treatment of /, m, n may be preferred in the process of the elimin- 
ation.] 

The wave-surface problem, in words, is this: Let the solid 
within any small volume of space round the origin of coordinates, 
0, be suddenly disturbed in any manner and then left to itself. 
It is required to find the surface at every point of which a pulse 
of disturbance is experienced at time t [t = 1 in the mathematical 
solution above]. 

[June 16, 1898. This problem I now find was stated very clearly 
and attacked with great analytical power by Blanchet in his 
" Me'moire sur la Propagation et la Polarisation du Mouvement 
dans tin milieu e'lastique inde"fini, cristallis d'une maniere quel- 
conque," the first of the three memoirs referred to in the footnote 
above. At the end of this paper he sums up his conclusion as 
follows : 

" 1. Dans un milieu 61astique, homogene, indfmi, cristal- 
lise" d'une maniere quelconque, le mouvement produit par un 
ebranlement central se propage par une onde plus ou moins 
compliqude dans sa forme. 



138 LECTURE XII. PART I. 

" 2. Pour chaque nappe de 1'onde, la vitesse de propagation 
est constante dans une meme direction, variable avec la direction 
suivant une loi qui depend de la forme de 1'onde. 

" 3. Pour une meme direction, les vitesses de vibration 
sont constamment paralleles entre elles dans une meme nappe 
de 1'onde pendant la duree du mouvement, et paralleles a des 
droites differentes pour les differentes nappes, ce qui constitue 
une veritable polarisation du mouvement." 

In 2 and 3 of this statement " direction " must be interpreted as 
meaning direction of the perpendicular to the tangent plane, and 
to 3 it is to be added that the three " droites differentes " mentioned 
in it are mutually perpendicular, because of the fifteen necessary 
equalities not assumed by Blanchet among the thirty-six coefficients. 
The second and third of Blanchet's memoirs (Liouville s Journal, 
Vol. VIL, 1842) are entitled " Memoire sur la Delimitation de 1'onde 
dans la propagation des Mouvements Vibratoires," and " Memoire 
sur une circonstarice remarquable de la Delimitation de 1'onde." 
They contain some exceedingly interesting conclusions, which 
Blanchet on the invitation of Liouville had worked out as ex- 
tensions to a crystallised body of results previously found by 
Poisson for an isotropic solid, regarding the space throughout 
which there is some movement of the elastic solid at any time 
after the cessation of the disturbing action within a small finite 
space. Cauchy had also worked on the same subject and had 
given an analytical method, his " Calcul des Residus," which 
Blanchet used with due acknowledgment. The two authors, 
working nearly simultaneously, seem to have found, each for him- 
self, all the main results, and each to have appreciated loyally the 
other's work. It is interesting also to find Poisson, Coriolis, Sturm, 
Cauchy, Liouville, and Duhamel reporting favourably and suggest- 
ively on Blanchet's memoirs, and Liouville helping him with advice 
in the course of his investigations. 

As part of his conclusion regarding " delimitation " Blanchet 
says, " II n'y a, en general, ni deplacement ni vitesse au-dela de 
la plus grande nappe des ondes " ; and Cauchy on the same 
subject in the Comptes Rendus, xiv. (1842), p. 13, excluding 
condensational-rarefactional waves, says, "Les deplacements et 
par suite les vitesses des molecules s'evanouiront par tous les 
points situes en dehors ou en dedans des deux ondes propagecs. 



CAUCHY ON BLANCHET. 139 

M Blanchet a remarque avec justesse qu'on ne pouvait, en Molar. 
gnral, en dire autant des points situe's entre les deux ondes. 
Toutefois il est bon d'observer que, meme en ces derniers points, 
les deplacements et les vitesses se reduisent a ze'ro quand on 
suppose nulle la dilatation du volume..., c'est a dire, en d'autres 
termes, quand les vibrations longitudinales disparaissent."*] 

Green treats the subject of waves in an aeolotropic elastic 
solid in a peculiar and most interesting manner for the purpose of 
forming a dynamical theory of " the propagation of light in 
crystallized media." He investigates conditions^ that "transverse 
vibrations shall always be accurately in the front of the wave," or, 
in modern language, that the wave may be purely distortional. 
He finds* 14 relations among his 21 coefficients by which this is 
secured for a double-sheeted wave-surface, which he finds to be 
identical with Fresnel's. There is necessarily a third sheet, 
although Green does not mention it at all. It is ellipsoidal, and 
corresponds essentially to a condensational-rarefactional wave with 
vibrations at every point perpendicular to the tangent plane. 
It is quite disconnected from the double-sheeted surface of the 
distortioual wave ; and a disturbing source can be so adjusted as 
to produce only distortional wave-motion with the double-sheeted 
wave-surface, or only the condensational-rarefactional wave-motion 
with the ellipsoidal wave-surface, or both kinds simultaneously. 
The three principal axes of the ellipsoidal wave-surface coincide 
with the three axes of symmetry found for the wave-surface of the 
distortional Fresnel-Green wave-motion. 

This dynamics of waves in an elastic solid is a fine subject for 
investigation, and I am sorry now to pass from it for a time. 

But if the war is to be directed to fighting down the difficulties 
which confront us in the undulatory theory of light it is not of 
the slightest use towards solving our difficulties, for us to have 
a medium which kindly permits distortional waves to be pro- 
pagated through it, even though it be aeolotropic. It is not 
enough to know that though the medium be aeolotropic it can let 
purely distortional waves through it, and that two out of the 

* These quotations are copied from a very interesting account of the work of 
Blanchet and Cauchy on this subject on pp. 627634 of Todhunter and Pearson's 
Elasticity, Vol. i. 

t Green's Mathematical Paper* : " On Propagation of Light in Crystallized 
Media," p. 293 : Reprint from Trans. Cambridge Philosophical Society, May 20, 1839. 

* Ibid. p. 309. 



140 LECTURE XII. PART I. 

three waves will be purely distortional. What we want is a 
medium which, when light is refracted and reflected, will under 
all circumstances give rise to distortional waves alone. Green's 
medium would fail in this respect when waves of light come to a 
surface of separation between two such mediums. All that Green 
secures is that there can be a purely distortional wave ; he does 
not secure that there shall not be a condensational wave. There 
would generally be condensational waves from the source. White- 
hot bodies, flames as of candles or gaslights, electric light of 
all kinds, would produce condensational waves, whether in an 
aeolotropic or isotropic medium, so far as Green's conditions here 
spoken of, go. What we want is a medium resisting condensation 
sufficiently ; a medium with an infinite or practically infinite 
bulk-modulus so great that the amount of energy, developed 
in the shape of condensational waves, has not been discovered 
by observation. 

As an essential in every reflection and refraction there may 
be a little loss of energy from the want of perfect polish in the 
surface, but as a rule, we have practically no loss of light in 
reflection and refraction at surfaces of glass and clear crystals. 
There perhaps is some but we have not discovered it. The 
medium that gives us the luminiferous vibrations must be such 
that if there is any part of the energy of the wave expended in 
condensational waves after refraction and reflection, the amount of 
it must be so small that it has not been discovered. Numerical 
observations have been made with great accuracy, in which, for 
example, Fresnel's formula for the ratio of normally incident and 

reflected light ( -r) is verified within closer than one per cent, 

I believe. Still a half per cent or a tenth per cent of the energy 
may for oblique incidences be converted into condensational waves, 
for all we know. But if any large percentage were converted into 
condensational waves, there would be a great deal of energy in 
condensational waves going about through space, and (to use for a 
moment an absurd mode of speaking of these things) there would 
be a " new force " that we know nothing of. There would be some 
tremendous action all through the universe produced by the energy 
of condensational waves if the energy of these were one-tenth, or 
one-hundredth per cent of the energy of the distortional waves. 
I believe that if in oblique reflection and refraction of light at 



QUESTION OF CONDENSATIONAL WAVES 141 

any surface, or in case of violent action in the source, there are Molar, 
condensational waves produced with anything like a thousandth 
or a ten-thousandth of the energy of the light and radiant heat 
which we know, we should have some prodigious effect, but which 
might, perhaps, have to be discovered by some other sense than 
we have. The want of indication of any such actions is sufficient 
to prove that if there are any in nature, they must be exceedingly 
small. But that there are such waves, I believe ; and I believe 
that the velocity of the unknown condensational wave that we are 
speaking of is the velocity of propagation of electro-static force. 

I say " believe " here in a somewhat guarded manner. I do 
not mean that I believe this as a matter of religious faith, but 
rather as a matter of strong scientific probability. If this is true 
of propagation of electro-static force, it is true that there is 
exceedingly little energy in the waves corresponding to the 
propagation of an electro-static force. That is however going 
beyond our tether of Molecular Dynamics. What I proposed 
in the introductory statement with reference to these Lectures 
was to bring what principles and results of the science of molecular 
dynamics I could enter upon, to bear upon the wave theory of 
light. We are sticking closely to that for the present, and we 
may say that we have nothing to do with condensational waves. 
Our medium is to be incompressible, and instead of Green's 
fourteen equations, we have merely one condition, that the 
medium is incompressible. It is obvious that this condition 
suffices to prevent the possibility of a wave of condensation at all 
and reduces our wave-surface to a surface with two sheets, like 
the Fresnel surface. But before passing away from that beautiful 
dynamical speculation of Green's, if we think of what the con- 
densational wave must be in an aeolotropic solid fulfilling Green's 
condition that it can have purely distortional waves proceeding in 
all directions the condition that two of the three waves which 
we investigated three-quarters of an hour ago shall be purely 
distortional I think we shall find also condensational waves, and 
that the wave-surfaces for them will be a set of concentric ellipsoids. 
It will be a single-sheeted surface, that is certain, because you 
have only one velocity corresponding to each tangent plane at the 
wave-surface. 

I shall now leave this subject for the present. We shall come 
back upon it again, perhaps, and look a little more into the 



142 LECTURE XII. PART T. 

question of moduluses of elasticity. We shall work up from an 
isotropic solid to the most general solid ; and we shall work down 
from the most general solid to an isotropic solid. We shall take 
first the most general value for the compressibility ; we shall then 
come to this subject again of assuming incompressibility. We 
shall then begin with the most general solid possible, and see what 
conditions we must impose to make it as symmetrical as is necessary 
for the Fresnel wave-surface. The molecular problem will prepare 
your way a good deal for this. 

I had intended to prepare something about the mass of the 
luminiferous ether. I have not had time to take it up, but I 
certainly shall do so before we have done with the subject. We 
shall go into the question of the density of the luminiferous ether, 
giving superior and inferior limits. We shall also consider what 
fraction of a gramme may be in one of these molecules and show 
what an enormously smaller fraction of a gramme we may suppose 
it to displace in the luminiferous ether. We shall try to get into 
the notion of this, that the molecule must be elastic and that there 
must be an enormous mass in its interior. Its outer part feels 
and touches the luminiferous ether. It is a very curious sup- 
position to make, of a molecular cavity lined with a rigid spherical 
shell ; but that something exists in the luminiferous ether and 
acts upon it in the manner that is faultily illustrated by our 
mechanical model, I absolutely believe. 

Just think of the effect of a shock consisting say of a collision 
between that and another molecule. Instead of its being broken 
into bits, let us suppose an unbroken spherical sheath around it. 
It will bound away, vibrating. Just imagine the central nucleus 
vibrating in one direction while the shell vibrates in the other, and 
you have a molecule with two parts going in opposite directions ; 
but differing from what I thought of the other day (Lecture IX. 
p. 96 above) in that one part is inside the other. The ether gets 
its motion from the outside part. Therefore I say that the most 
fundamental supposition we can make with reference to the origi- 
nating source of a sequence of waves of light is that illustrated 
by a globe vibrating to and fro in a straight line. 

We have already investigated (Lecture VIII. pp. 8689) the 
solution corresponding to that. Consider the spherical waves ; no 
vibrations for points in one certain diameter of the sphere ; maxi- 
mum vibrations in all points of the equatorial plane of that 



POLARISED SOURCE OF LIGHT. 143 

diameter and perpendicular to that plane ; for all points in the Molecular. 
quadrant of an arc of the spherical surface extending from axis to 
equator, vibrations in the plane of and tangential to the arc; and 
of magnitude proportional to the cosine of the latitude or angular 
distance from the equator and of intensity proportional to the 
square of the cosine of the latitude. Then in a wave travelling 
outwards, let the amplitude vary inversely as the distance from 
the centre, and therefore the intensity inversely as the square of 
the distance from the centre ; and you have a correct word-painting 
of the very simplest and most frequent sequence of vibrations 
constituting light. 

Ten minutes interval. 

Let us return to the consideration of the dynamics of refraction, 
ordinary dispersion, anomalous dispersion, and absorption. Begin- 
ning with our formulas as we left them yesterday (p. 120 above), 
let us consider what they become for the case r = oc ; that is to 
say, for static displacement of the containing shell. The inertia 
of the molecules will now not be called into play, and the case 
becomes simply that of the equilibrium of the set of springs whose 
stiffnesses we have denoted by c,, c,, ... , c,, c,-+i, when S, the end 
of Ci remote from IM,, is displaced through a space , and F, the 
end of Cj+i remote from nij t is fixed. 




Denoting by X the force with which S pulls and F resists, 
which, as inertia does not come into play, is therefore the force 
with which each spring is stretched, we have 

X = c^Xj = Cj (xj-i - xj) = ... = c, (ar, - *,) = c, ( - #,)...(!) ; 
or, as we may write it, 




whence 

11 11 

i + -+...+- + - 



i i i 

-+--4--+... +- + - 

C, C, C, Cj Cj+i 



144 LECTURE XII. PART IT. 

Molecular. Hence, unless at least one of c 1} c. 2 , ... , GJ +I is zero, ~ I is 

negative for r = <*> , and therefore ^ diminishes to oo as T- in- 
creases to + x> . Hence a large enough finite value of r makes 
= 0, and all larger values of T- make ^ negative. This is a 

particular, and an extreme, case of a very important result with 
which we shall have much to do later, in following the course of 
our formula when the period of the light is increased from any 
molecular period to the next greater molecular period ; and we 
may get quit of it for the present by assuming Cj +l to be zero. We 
may recover it again, and perceive its true physical significance, 
by supposing mj to be infinitely great ; and therefore we lose 
nothing of generality by taking c/ +1 = 0. This, in our formulas of 

to-day, makes X = and y = 1. The latter, by putting r = oo in 
our last formula of Lecture X. (p. 120 above), gives 

~R + K?R, + K jR tl + &c.) ...... (4). 



Subtracting this from our last formula of Lecture X. with 
T general, we find 



whence, by the preceding formula of Lecture X., (p. 120) 
1 p Cf-r- 

= 



A convenient modification of this formula is got by putting in it 



and Cl = ^ ...(8). 

/Cj 

Thus fi denotes the refractive index of the medium; and ^ 
denotes the period which m, would have as a vibrator, if the 
shell lining the ether were held fixed, and if the elastic connection 
between m, and interior masses were temporarily annulled. With 
these notations (6) becomes 



DYNAMICS OF DISPERSION. 145 

When the period of the light is very long in comparison with Molecular. 
the longest of the molecular periods of the embedded molecules, it 
is obvious that each material particle will be carried to and fro 
with almost exactly the same motion as the shell, in fact almost 
as if it were rigidly connected with the shell ; and therefore the 
velocity of light will be sensibly the same as if the masses of the 
particles were distributed homogeneously through the ether without 
any disturbance of its rigidity. 

Let us consider now how, when the period of the light is not 
infinitely long in comparison with any one of the molecular periods, 
the internal vibrations of the molecule modify the transmission 
of light through the medium. For simplicity at present let us 
suppose our molecule to have only one vibrating particle, the m^ of 
our formulas, which we will now call simply m, 
being the sum of the masses of the vibrators 
per unit volume of the ether. Imagine it con- 
nected with the shell or sheath, S, surrounding 
it, by massless springs (as in the accompanying 
diagram), through which it acts on the ether 
surrounding the sheath. Its influence on the 
transmission of light through the medium can 
be readily understood and calculated from the diagrams of p. 147, 
representing the well-known elementary dynamics of a pendulum 
vibrating in simple harmonic motion, when freely hung from a 
point S, (corresponding to the sheath lining our ideal cavity in 
ether), which is itself compelled by applied forces to move horizon- 
tally with simple harmonic motion. Figs. 1 and 2 illustrate the 
cases in which the period of the point of suspension, S, is longer 
than the period of the suspended pendulum with S fixed ; and 
figs. 3 and 4 cases in which it is shorter. In each diagram OM 
represents the length of an undisturbed simple pendulum whose 
period is equal to that of the motion of 8, the point of support of 
our disturbed pendulum SM. Thus if we denote the periods by 
r and * respectively, we have OM/SM = T*/**. Hence if and x 
denote respectively the simultaneous maximum displacements of 
S and M, we have 




If now for a moment we denote by w the mass of the single 
vibrating particle of our diagrams, the horizontal component of the 
T. L. 10 



146 LECTURE XII. PART IT. 

Moleculnr. pull exerted by the thread MS on >Sf at the instant of maximum 
displacement is w (2?r/T) 2 x ; and with m in place of w, we have 
the sum of the forces exerted by all the molecules per unit volume 
of the ether at the instant of maximum displacement of ether and 
molecules at any point. This force is subtracted from the re- 
stitutional force of the ether's elasticity, when the period of the 
light is greater than the molecular period (figs. 1 and 2) ; and is 
added to it, when the period of the light is less than the molecular 
period (figs. 3 and 4). 

Now if the rigidity of the ether be taken as unity, its elastic 
force per unit of volume at any point at the instant of its maximum 
displacement is (2?r/0 2 , where I is the wave-length of light in 
the ether with its embedded molecules. Hence the total actual 
institutional force on the ether is 

27T\ 2 /27T 



which must 



be equal to ( ) if the density of the ether be 

taken as unity. Hence dividing both members of our equation by 
(27r/r) 2 f, and denoting by the velocity of light through the 
medium consisting of ether and embedded molecules, and by /* its 
refractive index, we have 



whence by (10), 

''+'- ............ (m 



By quite analogous elementary dynamical considerations we find 



^ = 1 + m __ + w/ __ + - W|> ___ 

if, instead of one set of equal and similar molecules, there are 
several such sets, the molecules of one set differing from those 
of the others. The period *, or * or *, &c., of any one of the 
sets is simply the period of vibration of the interior mass, 
when the rigid spherical lining of the ether around it, which for 
brevity we shall henceforth call the sheath of the molecule, is 
held fixed. Instead of calling this sheath massless as hitherto 
we shall suppose it to have mass equal to that of the displaced 



DYNAMICS OF DISPERSION. 



147 



Molecular. 




Fig.1. 




102 



148 LECTURE XTI. PART IT. 

Molecular, ether ; so that if there were no interior masses the propagation 
of light would be sensibly the same as in homogeneous un- 
disturbed ether. The condition which we originally made, that 
the distance from molecule to molecule must be very great in 
comparison with the diameter of each molecule and very small 
in comparison with the wave-length, must be fulfilled in respect 
to the distances between molecules of different kinds. The 
number of molecules of each kind in a cube of edge equal to 
the wave-length must be very great ; and the numbers in any 
two such cubes must be equal, this last being the definition of 
homogeneity. These are substantially the conditions assumed by 
Sellmeier*, and our equation (13) is virtually identical with his 
original expression for the square of the refractive index. 

It is interesting to remark that our formula (9) for the effect 
on the velocity of regular periodic waves of light, of a multitude 
of equal and similar complex vibrating molecules embedded in 
the ether, shows that it is the same as the effect of as many 
sets of different simple vibrators as there are fundamental periods 
in our one complex vibrator : and that the masses of the equi- 
valent simple vibrators are given by the equations 

m^R ni^K*R, m^H^R,. 

m A ; ///. = , : //.,= ; &c....(14). 

P*l 4 P*, 4 pK, 4 

But although the formula for the velocity of regular periodic 
waves is the same, the distribution of energy, kinetic and potential, 
is essentially different on the two suppositions. This difference 
is of great importance in respect to absorption and fluorescence, 
as we shall see in considering these subjects later. 

[Added Sept. 23f, 1898. The dynamical theory of dispersion, 
as originally given by Sellmeier*, consisted in finding the velocity 
of light as affected by vibratory molecules embedded in ether, 

* Sellmeier, Pogg. Ann., Vol. 145, 1872, pp. 399, 520; Vol. 147, 1872, pp. 387, 
525. 

t [The substance of this was communicated to Sec. A of the British Association 
at Bristol on September 9, 1898, in two papers under the titles " The Dynamical 
Theory of Refraction, Dispersion, and Anomalous Dispersion," and " Continuity in 
Uudulatory Theory of Condensational-rarefactional Waves in Gases, Liquids, and 
Solids, of Distortional Waves in Solids, of Electric and Magnetic Waves in all 
Substances capable of transmitting them, and of Radiant Heat, Visible Light, 
Ultra-Violet Light and Rontgen Rays."] 



SELI.MEIER'S THEORY OF DISPERSION. 14-9 

such as those which had been suggested by Stokes* to account Molecular. 
for the dark lines of the solar spectrum. Sellmeier's mathematical 
work was founded on the simplest ideal of a molecular vibrator, 
which may be taken as a single material particle connected by a 
massless spring or springs with a rigid sheath lining a small vesicle 
in ether. He investigated the propagation of distortional waves, 
and found the following expression (which I give with slightly 
altered notation) for the square of the refractive index of light 
passing through ether studded with a very large number of 
vibratory molecules in every volume equal to the cube of the 
wave-length : 



where r denotes the period of the light ; *, *,, *, &c., the vibratory 
periods of the embedded molecules on the supposition of their 
sheaths held fixed ; and m, in,, m,,, &c., their masses. He showed 
that this formula agreed with all that was known in 1872 regard- 
ing ordinary dispersion, and that it contained what we cannot 
doubt is substantially the true dynamical explanation of anomalous 
dispersions, which had been discovered by Fox-Talbotf for the 
extraordinary ray in crystals of a chromium salt, by LerouxJ for 
iodine vapour, and by Christiansen for liquid solution of fuchsin, 
and had been experimentally investigated with great power by 
Kundt*. 

Sellmeier himself somewhat marred ff the physical value of his 
mathematical work by suggesting a distinction between refractive 
and absorptive molecules (" refractive und absorptive Theilchen "), 
and by seeming to confine the application of his formula to cases 
in which the longest of the molecular periods is small in com- 
parison with the period of the light. But the splendid value 
to physical science of his non-absorptive formula has been quite 
wonderfully proved by Rubens (who, however, inadvertently 



See Kirchhoff-Stokes-Thomson, Phil. Mag., March and July 1860. 
t Fox-Talbot, Proc. Roy. Soc. Edin., 187071. 
J Lertmx, Comptet rendiu, 55, 1862, pp. 126128. 

Christiansen, Ann. Phyt. Chtm., 141, 1870, pp. 479, 480; I'hil. Mag., 41, 
1871, p. 244 ; Annalet de Chimic, 25, 1872, pp. 213, 214. 
' Kondt, Pogg. Ann., Vols. 142, 143, 144, 146, 187172. 
tt Pogy. Ann., Vol. 147, 1H72, p. 525. 



150 LECTURE Xll. PART II. 

Molecular, quotes* it as if due to Ketteler). Fourteen years ago Langleyf 
had measured the refractivity of rock-salt for light and radiant 
heat of wave-lengths (in air or ether) from '43 of a mikrom to 
5'3 mikroms (the mikrom being 10" 6 of a metre, or 10~ 4 of a 
centimetre), and without measuring refractivities further, had 
measured wave-lengths as great as 15 mikroms in radiant heat. 
Within the last six years measurements of refractivity by Rubens, 
Paschen, and others, agreeing in a practically perfect way with 
Langley's through his range, have given us very accurate know- 
ledge of the refractivity of rock-salt and of sylvin (chloride of 
potassium) through the enormous range of from '4 of a mikrom to 
23 mikroms. 

Rubens began by using empirical and partly theoretical 
formulas which had been suggested by various theoretical and 
experimental writers, and obtained fairly accurate representations 
of the refractivities of flint-glass, quartz, fluorspar, sylvin, and 
rock-salt through ranges of wave-length from '4 to nearly 12 
mikroms. Two years later, further experiments extending the 
measure of refractivities of sylvin and rock-salt for light of wave- 
lengths up to 23 mikroms, showed deviations from the best of the 
previous empirical formulas increasing largely with increasing 

* Wied. Ann., Vol. 53, 1894, p. '267. In the formula quoted by llubens from 
Ketteler, substitute for /u x the value of fj. found by putting T x in Sellmeier's 
formula, and Ketteler's formula becomes identical with Sellmeier's. Remark that 
Ketteler's "M" is Sellmeier's " DIK-" according to my notation in the text. 

t Langley, Phil. Mag., 1886, 2nd half-year. 

[For a small unit of length Langley, fourteen years ago, used with great 
advantage and convenience the word "mikron" to denote the millionth of a 
metre. The letter n has no place in the metrical system, and I venture to suggest 
a change of spelling to "rnikrom" for the millionth of a metre, after the analogy 
of the English usage for millionths (mikrohm, mikro-ampere, mikrovolt). For a 
conveniently small corresponding unit of time I further venture to suggest 
" michron " to denote the period of vibration of light whose wave-length in ether 
is 1 mikrom. Thus, the velocity of light in ether being 8 x 10 8 metres per second, 
the michron is | x 10~ u of a second, and the velocity of light is 1 mikrom of space 
per michron of time. Thus the frequency of the highest ultra-violet light investi- 
gated by Schumann (-1 of a mikrom wave-length, Sitzttngsber. d. k. Gesellsclt. d. 
Wissensch. zu Wien, en. pp. 415 and 625, 1893) is 10 periods per michron of time. 
The period of sodium light (mean of lines D) is -58J32 of a michron ; the periods 
of the " Reststrahlen " of rock-salt and sylvin found by Rubens and Aschkinass 
(Wied. Ann. LXV. (1898), p. 241) are 51-2 and 61-1 michrons respectively. No 
practical inconvenience can ever arise from any possible confusion, or momentary 
forgetfulness, in respect to the similarity of sound between michrons of time and 
mikroms of space. K.] 

Rubens, Wied. Ann., Vols. 53, 54, 189495. 



RUBENS' RESULTS FOR ROCK-SALT AND SYLVIN. 151 

wave-lengths. Rubens then fell back* on the simple unmodified Molecular. 
Sellmeier formula, and found by it a practically perfect expression 
of the refractivities of those substances from '434 to 22'3 mikroms. 

And now for the splendid and really wonderful confirmation of 
the dynamical theory. One year later a paper by Rubens and 
Aschkinass'f describes experiments proving that radiant heat after 
five successive reflections from approximately parallel surfaces of 
rock-salt; ami again of sylvin ; is of mean wave-length 51'2 and 
61'1 mikroms respectively. The two formulas which Rubens had 
given in February 1897, as deduced solely from refractivities 
measured for wave-lengths of less than 23 mikroms, made /* 2 
negative for radiant heat of wave-lengths from 37 to 55 mikroms 
in the case of reflection from rock-salt, and of wave-lengths 
from 45 to 67 mikroms in the case of reflection from sylvin! 
(/*' J negative means that waves incident on the substance cannot 
enter it, but are totally reflected). 

These formulas, written with a somewhat important algebraic 
modification serving to identify them with Sellmeier's original 
expression, (13) above, and thus making their dynamical meaning 
clearer, are as follows : 

Rocket M - = 1-1875 + 1-1410 + 2-8504 



Sylvin ,,.- T 



- 4517-r 

The accompanying diagrams (pp. 152, 153) represent the squares of 
the refractive index of rock-salt and sylviu, calculated from these 
formulas through a range of periods from '434 of a michron to 
100 michrons. In each diagram the scale both of ordinate and 
abscissa from to 10 michrous is ten times the scale of the 
continuation from 10 to 100 michrons. Additions and sub- 
tractions to keep the ordinates within bounds are indicated on 
the diagrams. The circle and cross, on portion b of each curve, 
represent respectively the points where /i a = 1 and ft* 0. The 
critical periods exhibited in the formula are 
for rock-salt, 

1273 and 56 11, being the square roots of '01621 and 3149'3; 
for sylvin, 

1529 and 67'209, -0234 4517'1. 

* Kubens and Nichols, ll'itd. Ann., Vol. 60, 189697, p. 454. 

t Ruben* and Aschkinabs, Ifitd. Ann., Vol. 65, 1898, p. 241. 



152 



LECTURE XII. PART II. 



Molecular. 




?O 



RUBENS' RESULTS FOR ROCK-SALT AND SYLVIN. 153 




154 LECTURE XII. PART II. 

ir. The value of ^- passes from oo to + x as r rises through these 
values, and we have accordingly asymptotes at these points, 
represented by dotted ordinates in the diagrams. 

The agreement with observation is absolutely perfect through 
the whole range of Langley, Rubens, and Paschen from '4-S4 of a 
michron to 22'3 michrons. The observed refractivities are given 
by them to five significant figures, or four places of decimals ; and 
in a table of comparison given by Rubens and Nichols* the 
calculated numbers agree with the observed to the last place of 
decimals, both for rock-salt and sylvin. 

While in respect to refractivity there is this perfect agreement 
with Selltneier's formula through the range of periods from '434 
of a michron to fifty-one times this period (corresponding to 
nearly six octaves in music), it is to be remarked that with 
radiant heat of 22 3 michrons period, Rubens found, both for 
rock-salt and sylvin, so much absorption, increasing with increasing 
periods, as to prevent him from carrying on his measurements 
of refractivity to longer periods than 22 or 23 michrons, and to 
extinguish a large proportion of the radiant heat of 23 michrons 
period in its course through his prisms. Hence although Sellmeier's 
formula makes no allowance for the large absorptivity of the trans- 
parent medium, such as that thus proved in rock-salt and sylvin with 
radiant heat of periods less than half the critical period re in each 
case, it is satisfactory to know that large as it is the absorptivity 
produces very little effect on the velocity of propagation of radiant 
heat through the medium. This indeed is just what is to be 
expected from dynamical theory, which shows that the velocity 
of propagation is necessarily affected but very little by the forces 
which produce absorption, unless the absorptivity is so great that 
the intensity of a ray is almost annulled in travelling three or 
four times its wave-length through the medium. 

In an addition to a later lecture 1 intend to refer again to 
Helmholtz's introduction of resisting forces in simple proportion 
to velocities, by which he extended Sellmeier's formula to include 
true bodily absorption, and to recent modifications of the extended 
formula by himself and by Ketteler. Meantime it is interesting, 
and it may correct some misapprehension, to remark that so far 

* Wicd. Ann., Vol. 60, 18%~97, p. 454. 



THKEK KINDS OF CRITICAL PERIOD. 155 

as the dynamics of Sellmeier's single vibrating masses, or of my Moleeui 
complex molecule, goes, there is no necessity to expect any 
absorption at all, even for light or radiant heat of one of the chief 
critical periods *, as we shall see by the following general view of 
the circumstances of this and other critical periods. We shall see 
in fact that there are three kinds of critical period, 

(1) a period for which p* = 1, or velocity in the medium 
equal to velocity in pure undisturbed ether ; 

(2) a period for which /** = 0, or wave-velocity infinite in 
the medium ; 

(3) a period (any one of the /ic-periods) such that if we 
imagine passing through it with augmenting period, /** changes 
from w to 4- oo . 

The dynamical explanation of points where p.- = \ (marked 
by circles ou the curves), is simpler with my complex molecule 
than with Sellmeier's several sets of single vibrators. With my 
complex molecule it means a case in which No. 1 spring, that 
between //<, and the sheath, experiences no change of length. In 
order that this may be the case, the period of the light must be 
equal to some one of the free periods of the complex molecule, 
detached from the sheath (d temporarily annulled). With Sell- 
meier's arrangement, it is a case in which tendency to quicken 
the vibrations of the ether by one set of vibrators, is counter- 
balanced by tendency to slow them by another set, or other sets. 
It is a case which could not occur with only one set of equal and 
similar simple vibrators. 

The case we have next to consider, /A* = (or s = ), marked 
with crosses on the curve, occurs essentially just once for a single 
set of simple vibrators ; and it occurs as many times as there are 
sets of vibrators, if there are two or more sets with different 
periods; or, just once for each of the critical periods K, *,, *, &c. 
of my complex molecule. The dynamical explanation is particu- 
larly simple for a single set of Sellmeier vibrators ; it is the whole 
ether remaining undistorted and vibrating in one direction, while 
the masses m all vibrate in the opposite direction; the whole 
system being as it were two masses E and M connected by 
a single massless spring, and vibrating to and fro in opposite 
directions in a straight line with their common centre of inertia 



156 LECTURE XII. PART II. 

Molecular, at rest. Thus we see exactly the meaning of wave-velocity 
becoming infinite for the critical cases marked with a cross on 
the curves. 

Increase of the period beyond this critical value makes p- 
negative until we reach the next one of our critical values K, K,, 
K,,, &c., or the one of these, if there is only one. The meaning 
clearly is that light cannot penetrate the medium and is totally 
reflected from it wherever it falls on it. Thus for light or radiant 
heat of all periods corresponding to the interval between a point 
on one of our curves marked with a cross and the next asymptotic 
ordinate, the medium acts as silver does to visible light ; that 
is to say, it is impervious and gives total reflection. When we 
increase the period through one of our chief critical values /c, , , 
K, n &c., /*'-' passes from x to + =c . With exactly the critical 
period we have infinitely small velocity of propagation of light 
through the medium, and still total reflection of incident light. 
There would be infinitely great amplitude of the molecular 
vibrators, if the light could get into the medium ; but it cannot 
get in. With period just a little greater than one of these critical 
values, the reflection of the incident light is very nearly total ; 
the velocity of propagation of the light which enters is very 
small ; and the energy (kinetic and potential) of the molecular 
vibrators is very great in comparison with the energy (kinetic 
and potential) of the ether. 

Lastly, if there were no greater critical period than this which 
we have just passed, we should now have ordinary refraction with 
ordinary dispersion, at first large, becoming less and less with 
increased period, and /LI- diminishing asymptotically to the value 

-, where p denotes the density of the ether, and M the 



M + p 

sum of all the masses within the sheaths and connected to 
them by springs. But when we consider that the whole mass 
of the ponderable matter is embedded in the ether, and we 
cannot conceive of merely an infinitesimal portion of it clogging 
the ether in its luminiferous or electric vibrations, and that 
therefore for glass, or water, or even for rarefied air, M must be 
millions of millions of times p, we see how utterly our dynamical 
theory fails to carry us with any tolerable comfort, in trying to 
follow and understand the nature of waves and vibrations of 
periods longer than the 60 michrons (or 2 x 10~ 13 of a second) 



CONTINUITY IN UNDULATORY THEORY. 157 

touched by Rubens and Aschkinass*, up to 10~* of a second, or Molai 
further up to one thousandth of a second, or up to still longer 
periods. 

Yet we must try somehow to find and thoroughly understand 
continuity in the undulatory theory of condensational -rarefactional 
waves in gases, liquids, and solids, of distortional waves in solids, 
of electric and magnetic waves in all substances capable of trans- 
mitting them, and of radiant heat, visible light, ultra-violet light 
and Rb'ntgen rays. 

Consider the following three analogous cases : I. mechanical ; 
II. electrical ; III. electromagnetic. 

I. Imagine an ideally rigid globe of solid platinum of 12 
centim. diameter, hung inside an ideal rigid massless spherical 
shell of 13 centim. internal diameter, and of any convenient 
thickness. Let this shell be hung in air or under water by a very 
long cord, or let it be embedded in a great block of glass, or rock, 
or other elastic solid, electrically conductive or non-conductive, 
transparent or non-transparent for light. 

I. (1) By proper application of force between the shell and 
the nucleus cause the shell and nucleus to vibrate in opposite 
directions with simple harmonic motion through a relative total 
range of 10~* of a centimetre. We shall first suppose the shell 
to be in air. In this case, because of the small density of air 
compared with that of platinum, the relative total range will be 
practically that of the shell, and the nucleus may be considered 
as almost absolutely fixed. If the period is -fa of a second, 
(frequency 32 according to Lord Rayleigh's designation), a humming 
sound will be heard, certainly not excessively loud, but probably 
amply audible to an ear within a metre or half a metre of the 
shell. Increase the frequency to 256, and a very loud sound of 
the well-known musical character (C^) will be heard f. 

Increase the frequency now to 32 times this, that is to 8192 
periods per second, and an exceedingly loud note 5 octaves higher 
will be heard. It may be too loud a shriek to be tolerable ; if so, 

H'ied. Ann., Vol. 64, 1898. 

t Lord Rayleigh has found that with frequency 256, periodic condensation and 
rarefaction of the marvellously small amount 6 x 10~ of an atmosphere, or 
"addition and subtraction of densities far less than those to be found in our highest 
vacua," gives a perfectly audible sound. The amplitude of the aerial vibration, on 
each side of zero, corresponding to this, is 1-27 x 10~ 7 of a centimetre. Sound, Vol. n. 
p. 439 (2nd edition). 



158 LECTURE XTT. PART II. 

diminish the range till the sound is not too loud. Increase the 
frequency now successively according to the ratios of the diatonic 
scale, and the well-known musical notes will be each clearly and 
perfectly perceived through the whole of this octave. To some 
or all ears the musical notes will still be clear up to the G (24756 
periods per second) of the octave above, but we do not know 
from experience what kind of sound the ear would perceive 
for higher frequencies than 25000. We can scarcely believe 
that it would hear nothing, if the amplitude of the motion is 
suitable. 

To produce such relative motions of shell and nucleus as we 
have been considering, whether the shell is embedded in air, or 
water, or glass, or rock, or metal, a certain amount of work, not 
extravagantly great, must be done to supply the energy for the 
waves (both condensational and rarefactional), which are caused 
to proceed outwards in all directions. Suppose now, for example, 
we find how much work per second is required to maintain vibra- 
tion with a frequency of 1000 periods per second, through total 
relative motion of 10~ 3 of a centimetre. Keeping to the same 
rate of doing work, raise the frequency to 10 4 , 10 s , 10 6 , 10 9 , 10 12 , 
500 x 10 12 . We now hear nothing ; and we see nothing from any 
point of view in the line of the vibration of the centre of the 
shell which I shall call the axial line. But from all points of 
view, not in this line, we see a luminous point of homogeneous 
polarized yellow light, as it were in the centre of the shell, with 
increasing brilliance as we pass from any point of the axial line 
to the equatorial plane, keeping at equal distances from the centre. 
The line of vibration is everywhere in the meridional plane, and 
perpendicular to the line drawn to the centre. 

When the vibrating shell is surrounded by air, or water, or 
other fluid, and when the vibrations are of moderate frequency, or 
of anything less than a few hundred thousand periods per second, 
the waves proceeding outwards are condensational-rarefactional, 
with zero of alternate condensation and rarefaction at every point 
of the equatorial plane and maximum in the axial line. When 
the vibrating shell is embedded in an elastic solid extending to 
vast distances in all directions from it, two sets of waves, distor- 
tional and condensational-rarefactional, according respectively to 
the two descriptions which have been before us, proceed outwards 
with different velocities, that of the former essentially less than 



coNTiNrmr IN UNDULATORY THEORY. 159 

that of the latter in all known elastic solids*. Each of these Molar, 
propagational velocities is certainly independent of the frequency 
up to 10*, 10 s , or 10*, and probably up to any frequency not so 
high but that the wave-length is a large multiple of the distance 
from molecule to molecule of the solid. When we rise to fre- 
quencies of 4 x 10", 400 x 10" 800 x 10" and 3000 x 10", cor- 
responding to the already known range of long-period invisible 
radiant heat, of visible light, and of ultra-violet light, what 
becomes of the condensational-rarefactional waves which we have 
been considering ? How and about what range do we pass from 
the propagational velocities of 3 kilometres per second for distor- 
tional waves in glass, or 5 kilometres per second for the condensa- 
tional waves in glass, to the 200,000 kilometres per second for 
light in glass, and, perhaps, no condensational wave? Of one 
thing we may be quite sure ; the transition is continuous. Is it 
probable (if ether is absolutely incompressible, it is certainly 
possible) that the condensational-rarefactional wave becomes less 
and less with frequencies of from 10* to 4 x 10", and that there is 
absolutely none of it for periodic disturbances of frequencies of 
from 4 x 10" to 3000x10"? There is nothing unnatural or 
fruitlessly ideal in our ideal shell, and in giving it so high a 
frequency as the 500 x 10" of yellow light. It is absolutely 
certain that there is a definite dynamical theory for waves of 
light, to be enriched, not abolished, by electromagnetic theory; 
and it is interesting to find one certain line of transition from our 
distortional waves in glass, or metal, or rock, to our still better 
known waves of light. 

I. (2) Here is another still simpler transition from the dis- 
tortional waves in an elastic solid to waves of light. Still think 
of our massless rigid spherical shell, 13 centim. internal diameter, 
with our solid globe of platinum, 12 centim. diameter, hung in its 
interior. Instead of as formerly applying simple forces to produce 
contrary rectilinear vibrations of shell and nucleus, apply now 
a proper mutual forcive between shell and nucleus to give them 
oscillatory rotations in contrary directions. If the shell is hung 
in air or water, we should have a propagation outwards of dis- 
turbance due to viscosity, very interesting in itself; but we should 
have no motion that we know of appropriate to our present 
subject until we rise to frequencies of 10, 10 x 10", 400 x 10", 

* Math, ami Phy*. Papert, Vol. HI. Art. civ. p. 522, 



160 LECTURE XII. PART II. 

Molar. 800 x 10 12 , or 3000 x 10 12 , when we should have radiant heat, or 
visible light, or ultra-violet light proceeding from the outer surface 
of the shell, as it were from a point-source of light at the centre, 
with a character of polarization which we shall thoroughly con- 
sider a little later. But now let our massless shell be embedded 
far in the interior of a vast mass of glass, or metal, or rock, or of 
any homogeneous elastic solid, firmly attached to it all round, so 
that neither splitting away nor tangential slip shall be possible. 
Purely distortional waves will spread out in all directions except 
the axial. Suppose, to fix our ideas, we begin with vibrations of 
one-second period, and let the elastic solid be either glass or iron. 
At distances of hundreds of kilometres (that is to say, distances 
great in comparison with the wave-length and great in comparison 
with the radius of the shell), the wave-length will be approxi- 
mately 3 kilometres*. Increase the frequency now to 1000 
periods per second : at distances of hundreds of metres the wave- 
length will be about 3 metres. Increase now the frequency to 
10 6 periods per second ; the wave-length will be 3 millim., and 
this not only at distances of several times the radius of the shell, 
but throughout the elastic medium from close to the outer surface 
of the shell; because the wave-length now is a small fraction 
of the radius of the shell. Increase the frequency further to 
1000 x 10 B periods per second ; the wave-length will be 3 x 10~ 3 of 
a millim., or 3 rnikroms, if, as in all probability is the case, the 
distance between the centres of contiguous molecules in glass and 
in iron is less than a five-hundredth of a mikrom. But it is 
probable that the distance between centres of contiguous molecules 
in glass and in iron is greater than 10~ 5 of a mikrom, and there- 
fore it is probable that neither of these solids can transmit waves 
of distortional motion of their own ponderable matter, of so short 
a wave-length as 10~ 5 of a mikrom. Hence it is probable that if 
we increase the frequency of the rotational vibrations of our shell 
to one hundred thousand times 1000 x 10", that is to say, to 
100 x 10 12 , no distortional wave of motion of the ponderable 
matter can be transmitted outwards ; but it seems quite certain 
that distortional waves of radiant heat in ether will be produced 
close to the boundary of the vibrating shell, although it is also 
probable that if the surrounding solid is either glass or iron, 
these waves will not be transmitted far outwards, but will be 

* Math, and Phi/x. Papers, Vol. in. Art. civ. p. 522. 



CONTINUITY OF UNDULATORY THEORY. 161 

absorbed, that is to say converted into non-undulatory thermal Molar, 
motions, within a few mikroms of their origin. 

Lastly, suppose the elastic solid around our oscillating shell 
to be a concentric spherical shell of homogeneous glass of a few 
centimetres, or a few metres, thickness and of refractive index 
1-5 for D light. Let the frequency of the oscillations be increased 
to 5*092 x 10" periods per second, or its period reduced to -58932 
of a michron : homogeneous yellow light of period equal to the 
mean of the periods of the two sodium lines will be propagated 
outwards through the glass with wave-length of about f x '58932 
of a mikrom, and out from the glass into air with wave-length 
of "58932 of a mikrom. The light will be of maximum intensity 
in the equatorial plane and zero in either direction along the axis, 
and its plane of polarization will be everywhere the meridional 
plane. It is interesting to remark that the axis of rotation of the 
ether for this case coincides everywhere with the line of vibration 
of the ether in the case first considered ; that is to say, in the 
case in which the shell vibrated to and fro in a straight line, 
instead of, as in the second case, rotating through an infinitesimal 
angle round the same line. 

A full mathematical investigation of the motion of the elastic 
medium at all distances from the originating shell, for each of the 
cases of I. (1) and I. (2), will be found later (p. 190) in these 
lectures. 

II. An electrical analogy for I. ( 1 ) is presented by sub- 
stituting for our massless shell an ideally rigid, infinitely massive 
shell of glass or other non-conductor of electricity, and for our 
massive platinum nucleus a massless non-conducting globe elec- 
trified with a given quantity of electricity. For simplicity we 
shall suppose our apparatus to be surrounded by air or ether. 
Vibrations to and fro in a straight line are to be maintained by 
force between shell and nucleus as in I. (1). Or, consider simply 
a fixed solid non-conducting globe coated with two circular caps of 
metal, leaving an equatorial non-conducting zone between them, 
and let thin wires from a distant alternate-current dynamo, or 
electrostatic inductor, give periodically varying opposite electri- 
fications to the two caps. For moderate frequencies we have 
a periodic variation of electrostatic force in the air or ether sur- 
rounding the apparatus, which we can readily follow in imagination, 
and can measure by proper electrostatic measuring apparatus. 
T. L. 11 



162 LECTURE XII. PART II. 

Its phase, with moderate frequencies, is very exactly the same 
as that of the electric vibrator. Now suppose the frequency of 
the vibrator to be raised to several hundred million million 
periods per second. We shall have polarized light proceeding as 
if from an ideal point-source at the centre of the vibrator and 
answering fully to the description of I. (1). Does the phase of 
variation of the electrostatic force in the axial line outside the 
apparatus remain exactly the same as that of the vibrator ? An 
affirmative answer to this question would mean that the velocity 
of propagation of electrostatic force is infinite. A negative 
answer would mean that there is a finite velocity of propagation 
for electrostatic force. 

III. The shell and interior electrified non-conducting massless 
globe being the same as in II., let now a forcive be applied 
between shell and nucleus to produce rotational oscillations as 
in I. (2). When the frequency of the oscillations is moderate, 
there will be no alteration of the electrostatic force and no 
perceptible magnetic force in the air or ether around our ap- 
paratus. Let now the frequency be raised to several hundred 
million million periods per second ; we shall have visible polarized 
light proceeding as if from an ideal point-source at the centre 
and answering fully to the description of the light of I. (2). 
The same result would be obtained by taking simply a fixed 
solid non-conducting globe and laying on wire on its surface 
approximately along the circumferences of equidistant circles of 
latitude, and, by the use of a distant source (as in II.), sending 
an alternate current through this wire. In this case, while there 
is no manifestation of electrostatic force, there is strong alter- 
nating magnetic force, which in the space outside the globe is 
as if from an ideal infinitesimal magnet with alternating magneti- 
zation, placed at the centre of the globe and with its magnetic 
axis in our axial line.] 



LECTURE XIII. 
SATURDAY, October 11, 8 P.M. 



PROF. MORLEY has already partially solved the definite dyna- Molecular. 
mical problem that I proposed to you last Wednesday (p. 103 
above) so far as determining four of the fundamental periods; 
and you may be interested in knowing the result. He finds roots, 
*-*, *,-, &c., = 3-46, I'OOo, -298, '087; each root not being very 
different from three times the next after it. I will not go into the 
affair any further just now. I just wish to call your attention to what 
Prof. Morley has already done upon the example that I suggested 
for our arithmetical laboratory. I think it will be worth while 
also to work out the energy ratios* (p. 74). In selecting this 
example, I designed a case for which the arithmetic would of 
necessity be highly convergent. But I chose it primarily because 
it is something like the kind of thing that presents itself in 
the true molecule : An elastic complex molecule consisting of a 
finite number of discontinuous masses elastically connected (with 
enormous masses in the central parts, that seems certain): the 
whole embedded in the ether and acted on by the ether in virtue 
of elastic connections which, unless the molecule were rigid and 
embedded in the ether simply like a rigid mass embedded in jelly, 
must consist of elastic bonds analogous to springs. 

I think you will be interested in looking at this model which, 
by the kindness of Prof. Rowland, I am now able to show you. 
It is made on the same plan as a wave machine which I made 
many years ago for use in my Glasgow University classes, and 
finally modified in preparations for a lecture given to the Royal 

* This was done by Mr Morley who kindly gave his results to the "Coeffi- 
cients " on the 17th Oct. See Lecture XIX. below. 

112 



164 



LECTURE XIII. 



Molecular. Institution about two years ago on " The Size of Atoms*." I 
think those who are interested in the illustration of dynamical 





problems will find this a very nice and convenient method. If 
you will look at it, you will see how the thing is done : Pianoforte 
wire, bent around three pins in the way you 
see here, supports each bar. These pins are 
slanted in such a way as to cause the wire 
to press in close to the bar so as to hold it 
quite firm. The wood is slightly cut away 

to prevent the wire from touching it above and below the pins, 
so that there may be no impairment of elastic action due to slip of 
steel on wood. The wire used is fine steel pianoforte wire ; that 
is the most elastic substance available, and it seems to me, indeed, 
by far the most elastic of all the materials known to us [except 
crystals; Jan. 19, 1899]. A heavy weight is hung on the lower 
end of the wire to keep it tightly stretched. 

Prof. Rowland is going to have another machine made, which 
I think you will be pleased with a continuous wave machine. 
This of mine is not a wave machine, but a machine for illus- 
trating the vibrations of a finite group of several elastically 
connected particles. The connecting springs are represented by 
the torsional springiness of the three portions of connecting wire 
and the fourth portion by which the upper mass is hung. In this 
case gravity contributes nothing to the effect except to stretch 
the wire. If we stretch the wire between two sides of a portable 

* February 3, 1883 ; Proc. Royal Institution, Vol. x. p. 185 ; Popular Lectures 
and Addresses, Vol. i. p. 154. 



JEOLOTROPY OF ELASTIC QUALITY. 165 

frame, we might take our model to an ideal lecture-room at the Molecular, 
centre of the earth, and it would work exactly as you see it working 
now. You will understand that these upper masses correspond 
to w,, ?n,, m,. In all we have four masses here, of which the 
lowest represents the spherical shell lining the ether around 
our ideal cavity. I will just apply a moving force to this lower 
mass, P. To realize the circumstances of our case more fully, we 
should have a spring connected with a vibrator to pull P with, 
and perhaps we may get that up before the next lecture. [Done 
by Professor Rowland; see Lecture XIV.] I shall attempt no 
more at present than to cause this first particle to move to and 
fro in a period perceptibly shorter than the shortest of the three 
fundamental periods which we have when the lowest bar is held 
fixed. The result is scarcely sensible motion of the others. 
I do not know that there would be any sensible motion at all if 
I had observed to keep the greatest range of this lowest particle 
to its original extent on the two sides of its mean position. 

The first part of our lecture this evening I propose to be a 
continuation of our conference regarding aeolotropy. The second 
part will be molecular dynamics. I propose to look at this question 
a little, but I want to look very particularly to some of the points 
connected with the conceivable circumstances by which we can 
account for not merely regular refraction, but anomalous dispersion, 
and both the absorption that we have in liquids and very opaque 
bodies and such absorption as is demonstrated by the very fine 
dark lines of the solar spectrum which are now shown more 
splendidly than ever by Prof. Rowland's gratings. 

I shall speak now of aeolotropy. The equations by which Molar. 
Green realized the condition that two of the three waves having 
fronts parallel to one plane shall be distortional, are in this respect 
equivalent to a very easily understood condition that I may 
illustrate first by considering the more general problem. That 
problem is similar to another of the very greatest simplicity, 
which is the well-known problem of the displacement of a particle 
subject to forces acting upon it in different directions from fixed 
centres. An infinitesimal displacement in any direction being 
considered, the question is, when is the return force in the 
direction of the displacement? As we know, there are three 
directions at right angles to one another, in which the return 
force is in the direction of the displacement. The sole difference 



166 LECTURE XIII. 

between that very trite problem and our problem of yesterday 
(Lecture XII. p. 135), is that in yesterday's the question is 
put with reference to a whole infinite plane in an infinite 
homogeneous solid, which is displaced in any direction between 
two fixed parallel planes on its two sides, to which it is always 
kept parallel. Considering force per unit of area, we have the 
same question, when is the return force in the direction of the 
displacement ? And the answer is, there are three directions 
at right angles to one another in which the return force is in 
the direction of the displacement. Those three directions are 
generally oblique to the plane ; but Green found the conditions 
under which one will be perpendicular to the plane, and the 
other two in the plane. 

I shall now enter upon the subject more practically in 
respect to the application to the wave theory of light, and begin 
by preparing to introduce the condition of incompressibility. 
Take first the well-known equations of motion for an isotropic 
solid and express in them the condition that the body is incom- 
pressible. The equations are : 






Suppose now the resistance to compression is infinite, which 
means, make k = oc at the same time that we have 8 = 0. What 
then is to become of the first term of the second members of 
these equations? We simply take (k+$n)S=p, and write the 

second members -J? + 7l V 2 f, &c., accordingly. This requires no 

hypothesis whatever. We may now take k = oo , 8 = 0, without 
interfering with the form of our equations. These equations, 
without any condition whatever as to , 7;, with the condition 
p = (k + ^ri) 8, are the equations necessary and sufficient for the 
problem. On the other hand, if k = oo , the condition that that 
involves is 



FRENCH CLASSICAL NOMENCLATURE. 167 

which gives four equations in all for the four unknown quantities Molar. 

fc if, fc P- 

Precisely the same thing may be done in respect to equations 
(1), (2), (3) of p. 132 above for a solid with 21 independent 
coefficients. We will have this equation, 8 = 0, again for an 
aeolotropic body, and a corresponding equality to infinity. I am 
not going to introduce any of these formulas at present. In 
the meantime, I tell you a principle that is obvious. In 

order to introduce the condition -j^ + -? + -r = into our 

dx ay dz 

general equation of energy with its 21 coefficients, which involves 
a quadratic expression in terms of the six quantities that we 
have denoted by e, /, g, a, 6, c, we must modify the quadratic 
into a form in which we have (e +f+ g) 3 with a coefficient That 
coefficient equated to infinity, and e+f+g = 0, leave us the 
general equations of equilibrium of an incompressible aeolotropic 
elastic solid. 

I want to call your attention to the kind of deviation from 
isotropy which is annulled by Green's equations among the 
coefficients expressing that two out of the three waves shall be 
purely distortional, and the third shall be condensational-rare- 
factional. 

The next thing to an isotropic body is one possessing what 
Rankine calls cyboid asymmetry. Raukine marks an era in 
philology and scientific nomenclature. In England, and I 
believe in America also, there has been a classical reaction, 
or reform, according to which, instead of taking all our 
Greek words through the French mill changing K (kappa) into 
c, and v (upsilon) into y, we spell in English, and we pronounce 
Greek words, and even some Latin words, more nearly according 
to what we may imagine to be the actual usage of the ancients. 
We cannot however in the present generation get over Kuros 
instead of Cyrus, Kikero instead of Cicero. Rankine is a curious 
specimen of the very last of the French classical style. Rankine 
was the last writer to speak of cinematics instead of kinematics. 
Cyboid, which he uses, is a very good word, but I do not know 
that there is any need of introducing it instead of Cubic. Cubic 
is an exception to the older classical derivation in that u is not 
changed into y; it should be cybic, and cube should be cybe 
(I suppose Kv/3o<; to be the Greek word). Cyboid obviously means 



168 LECTURE XIII. 

Molar. cube-like, or cubic, and it is taken from the Greek in Rankine's 
manner, now old-fashioned. 

Rankine gives the equations that will leave cubic asymmetry. 
He afterwards makes the very apposite remark that Sir David 
Brewster discovered that kind of variation from isotropy in 
analcime. I only came to this in Rankine two or three days ago. 
But I remember going through the same thing myself not long 
ago, and I said to Stokes (I always consult my great authority 
Stokes whenever I get a chance) " Surely there may be some- 
thing found in nature to exemplify this kind of asymmetry ; 
would it not be likely to be found in crystals of the cubic class ?" 
Stokes he knows almost everything instantly said. " Sir David 
Brewster thought he had found it in cubic crystals, but there 
is another explanation ; it may be owing to the effect of the 
cleavage planes, or the separation of the crystal into several 
crystalline laminas" I do not remember all that Stokes said, 
but he distinctly denied that Brewster's experiment showed a true 
instance of cubic optical asymmetry. He pointed out that an 
exceedingly slight deviation from cubic isotropy would show 
very markedly on elementary phenomena of light, and might be 
very readily tested by means of ordinary optical instruments. 
The fact that nothing of the kind has been discovered is 
absolute evidence that the deviation, if there is any, from optic 
isotropy in a crystal of the cubic class, is exceedingly small in 
comparison with the deviation from isotropy presented by ordinary 
doubly refracting crystals. 

Molecular. As a matter of fact, square a3olotropy is found in a pocket 
handkerchief or piece of square-woven cloth, supposing the warp 
and woof to be accurately similar, a supposition that does not 
hold of ordinary cloth. Take wire- cloth carefully made in 
squares; that is symmetrical and equal in its moduluses with 
reference to two axes at right angles to one another. There 
will be a vast difference according as you pull out one side 
and compress the other, or pull out one diagonal and compress the 
other. Take the extreme case of a cloth woven up with inex- 
tensible frictionless threads, and there is an absolute resistance to 
distortion in two directions at right angles to one another, and 
no resistance at all to distortion of the kind that is presented 
in changing it from square to rhombic shape. That is to say, a 
framework of this kind has no resistance to shearing parallel 



RANKINE'S NOMENCLATURE. 



169 





to the sides ; in other words, to the distortion produced Moleculi 
by lengthening one diagonal and shortening the 
other. Now imagine cut out of this pocket hand- 
kerchief, a square with sides parallel to the dia- 
gonals, making a pattern of this kind. There is 
a square that has infinite resistance to shearing parallel to its 
sides, and zero resistance to pulling out in the 
direction perpendicular to either pair of sides. This 
is not altogether a trivial illustration. Surgeons 
make use of it in their bandages. A person not 
familiar with the theory of elastic solids might cut a strip of 
cloth parallel to warp or woof; but cut it obliquely and you have 
a conveniently pliable and longitudinally semielastic character 
that allows it to serve for some kinds of bandage. 

Imagine an elastic solid made up in that kind of way, with 
that kind of deviation from isotropy ; and you have clearly two 
different rigidities for different distortions in the same plane. 
I remember that Rankine in one of his early papers proved this 
to be impossible ! He proved a proposition to the effect that 
the rigidity was the same for all distortions in the same plane. 
That was no doubt founded on some special supposition as to 
arrangement of molecules and may be true for the particular 
arrangement assumed ; but it is clearly fallacious in respect to 
true elastic solids. Rankine in his first paper made too short work 
of the elastic solid in respect to possibilities of aujlotropy. He 
soon after took it up very much on the same foundation as Green 
with his 21 coefficients, but still under the bondage of his old 
proposition that rigidity is the same for all distortions in the 
same plane. Yet a little later he escaped from the yoke, and 
took his revenge splendidly by giving a fine Greek name " cyboid 
asymmetry" to designate the special crystalline quality of which 
he had proved the impossibility ! 

I must read to you some of the fine words that Rankine has Molar, 
introduced into science in his work on the elasticity of solids. 
That is really the first place I know of, except in Green, in 
which this thing has been gone into thoroughly. It is not 
really satisfactory in Rankine except in the way in which he 
carries out the algebra of the subject, and the determinants 
and matrices that he goes into so very finely. But what I want 
to call attention to now is his grand names. I do not know 



170 LECTURE Xlll. 

whether Prof. Sylvester ever looked at these names ; I think he 
would be rather pleased with them. " Thlipsinomic transforma- 
tions," " Umbral surfaces," and so on. Any one who will learn the 
meaning of all these words will obtain a large mass of knowledge 
with respect to an elastic solid. The simple, good words, " strain 
and stress," are due to Rankine ; " potential " energy also. Hear 
also the grand words " Thlipsinomic, Tasinomic, Platythliptic, 
Euthytatic, Metatatic, Heterotatic, Plagiotatic, Orthotatic, Pan- 
tatic, Cybotatic, Goniothliptic, Euthythliptic," &c. 

You may now understand what cyboid asymmetry is, or as I 
prefer to call it, cubic asolotropy. Rankine had not the word 
aBolotropy; that came in from myself* later. Cyboid or cubic 
seolotropy is the kind of asolotropy exhibited by a cubic grating ; 
as it were a structure built up of uniform cubic frames. There 
[Feb. 14, 1899 ; a skeleton cube, of twelve equal wooden rods with 
their ends fixed in eight india-rubber balls forming its corners] 
is a thing that would be isotropic, except for its smaller rigidity 
for one than for the other of the two principal distortions in each 
one of the planes of symmetry. 

I will go no further into that just now ; but I hope that in 
the next lecture, or somehow before we have ended, we may be 
able to face the problem of introducing the relations among the 
21 moduluses which are sufficient to do away with all obliquities 
with reference to three rectangular axes. But you can do this in 
a moment equate to zero enough of the 21 coefficients to fulfil 
two conditions, (1) that if you compress a cube of the body by 
three balancing pairs of pressures, equal or unequal, perpendicular 
to its three pairs of faces, it will remain rectangular, and (2) that 
if you apply, in four planes meeting in four parallel edges, balanc- 
ing tangential forces perpendicular to these edges, the angles at 
these edges will be made alternately acute and obtuse, and the 
angles at the other eight edges will remain right angles. [Feb. 14, 
1899 ; here are the required annulments of coefficients to fulfil 
those two conditions, with OX, OY, OZ taken parallel to edges of 
the undisturbed cube, and with the notation of Lecture II. p. 23 : 

42 = 0; 43 = 0; 51=0; 53 = 0; 61=0; 02 = 0{ 
41=0; 52 = 0; 63 = 0; 56 = ; 64 = ; 45 = 0j'" ( '' 



* See p. 118 above. 



JEOLOTROP1C ELASTICITY WITHOUT SKEWNESS. l7l 

Thus the formula for potential energy becomes reduced to Molar. 

12ef) 

6 + 66c 8 (2). 

With these nine coefficients, 11, 22, 33; 23, 31, 12; 44, 55, 
66 ; all independent ; the elastic solid would present two different 
rigidities for the two distortions of each of the planes (yz\ (zx), (xy\ 
one by shearing motion parallel to either of the two other planes, 
the other by shearing motion parallel to either of the planes 
bisecting the right angles between those planes. The values of 
these rigidities are as follows for the cases of shearing motions 
parallel, and at 45, to our principal planes : 



Distortion 


Plane of Distortion 


Line of Motion 


Rigidity 


a 
f-9 


(y*) 


y or z 
45 to y and z 


44 

|{$ (22 + 33) -23} 


b 




t or x 


55 


9- 




45 to z and x 


itt(33 + ll)-31} 


e 


M 


x or y 
45 to ./ and y 


66 



From the fact that squares only of a, b, c appear in the equa- 
tion of energy, we see that in equilibrium* these distortions are 
separately balanced by the tangential stresses S, T, U. And we 
conclude that there can be plane waves of purely longitudinal 
motion (condensational-rarefactional) and waves of pure distortion 
travelling in the directions x, y, z with their fronts perpendicular 
to these directions ; and that their velocities f are as shown in the 
following table : 

Lecture U. p. 24. 

t Equations (1) above, with equations (4), (7), (8) of Lecture XI. p. 133, applied 
respectively to the cases 

m = 0, n = 0, = 0; n=0, J = 0, ?=0; 1=0, m=0, f=0. 



172 



LECTURE XIII. 



Molar. 



Wave-front 


Line of Vibration 


Quality 


Velocity 




fl 


condensational-rarefactional 


\/7 


(y*) 


y 


purely distortional 


N/f 




Z 





755 

V 7 




y 


coudensational-rarefactional 


722 

V7 


(MT) 


z 


purely distortional 


\/7 




X 


it 11 


x/? 




z 


condensational-rarefactional 


V? 


M 


X 


purely distortional 


/55 

V 7 




y 





N/7 



Selecting from these the purely distortional waves, and taking 
them in pairs having equal velocities, we have the following 
convenient table : 



Wave-front 


Line of Vibration 


Plane of Distortion 


Velocity 


(^) 
() 


y 


(y) 


^ 


0/2) 

(y*) 


z 

X 


(?) 


755 

V7 


(0 

(^) 


X 

y 


(*y) 


766 

V7 



EQUALISATION OF VELOCITIES FOR ONE PLANE OF DISTORTION. 173 

In this table we find one and the same velocity, ^44/p, for two Molar. 
different waves with fronts parallel to x, and having their lines of 
vibration parallel to y and z respectively, and therefore each having 
yz for its plane of distortion. If now we apply the formulas of 
Lecture XL, p. 133, to investigate the velocities of other waves 
having the same plane of distortion for instance, waves with 
fronts parallel to x and lines of vibration at 45 to y and z, we 
find different propagational velocities unless 



which makes them all equal. Thus, and by similar considerations 
relatively to y and z, we see that each of the three propagational 
velocities given in the preceding table is the same for all waves 
having the same plane of distortion, if the following conditions 
are fulfilled : 



55 =${(33 + 11) -31} .................. (3). 

66 = if* (11 + 22) -12}j 

These three equations simply express the condition that in 
each of the three coordinate planes the rigidity due to a shear 
parallel to either of the two other coordinate planes is equal to 
the rigidity due to a shear parallel to either plane bisecting the 
right angle between them. 

Greenf found fourteen equations among his 21 coefficients to 
express that there can be a purely distortional wave with wave- 
front in any plane whatever. Three more equations^ express 
further that the planes chosen for the coordinates are planes 
of symmetry. The conditions which we have considered have 
given us, in (1) and (3) above, fifteen equations which are identical 
with fifteen of Green's. His other two are 

11=22 = 33 .......................... (4). 

With all these seventeen equations among the coefficients, the 
equation of energy becomes reduced to 



Compare with formula n = J (3 - B) in Lecture II. p. 25, which is the condition 
that the rigidity ia the same for all distortions in any one of the three coordinate 
planes for the case of 11 = 22 = 33, there considered. 

t Collected Papert, p. 309. 

J Ibid., pp. 303, 309. 



174 



LECTURE XIII. 



It is easy now to verify Green's result. Remember that if we 
denote by 77, infinitesimal displacements of a point of the 
solid from its undisturbed position (x, y, 2), we have 



_ d% dr) j _d][ d _ drj 
dy dz ' dz dx ' dx 

Denoting now by 1 1 1 dxdydz integration throughout a volume 

F, with the condition that , 77, f are each zero at every point of 
the boundary, we find by a well-known method of double integra- 
tion by parts*, 

dr, dP 
dz dy 



/// 



*** 
///"** g-///-** I 

///*** i 3-//H 

And by aid of this transformation we find 



I 



.(7). 



fff 7 

dxdyd 
JJJ 



,, 

4 



- -,- 
dy dz 



../rfi ^r\ 2 ^/^ ^vi 

+ 55 (-/--/ + 66 ^ - ~ H 
\dz dx) \dx dyj } 

............ (8). 



Let now V be the space between two infinite planes, paralle 1 
to the fronts of any series of purely distortional waves traversing 
the space between them, these planes being taken at places at 
which for an instant the displacement is zero. They may, for 
instance, be the planes, half a wave-length asunder, of two 
consecutive zeros of displacement in a series of simple harmonic 
waves. Let P be any plane of the vibrating solid parallel to 
them ; let p be its distance from the origin of coordinates ; and 

q, its displacement at any instant. Then ^ is the molecular 



* Compare Math, and Phys. Papers, Art. xcix. Vol. in. p. 448 ; and "On the 
Kcflexion and Refraction of Light," Phil. Hag. 1888, 2nd half-year. 



GREEN'S DYNAMICS FOR FRESNEL'S WAVE-SURFACE. 175 

rotation of the solid infinitely near to this plane on each side ; and Molar. 
by the rectangular resolution of rotations, we have 

tdq^^d*. m dq = d_dS. n &_*?_# , 9) 
dp dy dz' dp dz da;' dp dx dy " 

where I, m, n denote the direction cosines of the line in P perpen- 
dicular to the direction of q, this being the axis of the molecular 
rotation. Using these new values in (8), and remarking that the 
first chief term vanishes because the displacement is purely 
distortional, we have 

2 fjfdxdydz E = (44/ 1 + 55w + 66n') fffdxdyd* (^Y. . .(10). 

Hence we see that no condensation or rarefaction accompanies 
our plane distortional waves, and that their velocity of propagation 
is 



( .11 ) 



This is Fresnel's formula for the propagational velocity of a 
plane wave in a crystal with (/, m, n) denoting the direction of 
vibration ; while in Green's theory (/, m, n) is the line in the 
wave-front perpendicular to the direction of vibration. The 
wave-surface is identical with Fresnel's.] 

I will read to you Green's own statement of the relative 
tactics of the motion in his and in Fresnel's waves. Here it 
is at the bottom of page 304- of Green's Collected Papei's: " We 
" thus see that if we conceive a section made in the ellipsoid 
"to which the equation (10) belongs, by a plane passing through 
" its centre and parallel to the wave's front, this section, when 
" turned 90 degrees in its own plane, will coincide with a similar 
" section of the ellipsoid to which the equation (8) belongs, and 
" which gives the directions of the disturbance that will cause 
" a plane wave to propagate itself without subdivision, and the 
"velocity of propagation parallel to its own front. The change 
" of position here made in the elliptical section is evidently 
" equivalent to supposing the actual disturbances of the ethereal 
" particles to be parallel to the plane usually denominated as the 
"plane of polarization." 

Before we separate this evening, return for a few minutes 
to our problem of vibratory molecules embedded in an elastic 



176 LECTURE XIII. 

Molecular, solid ; and let us consider particularly the application of this 
dynamical theory to the Fraimhofer double dark line D of sodium- 
vapour. 

[March 1, 1899.* For a perfectly definite mechanical repre- 
sentation of Sellmeier's theory, imagine for each molecule of 
sodium-vapour a spherical hollow in ether, lined with a thin rigid 
spherical shell, of mass equal to the mass of homogeneous ether 
which would fill the hollow. This rigid lining of the hollow we 
shall call the sheath of the molecule, or briefly the sheath. 
Within this put two rigid spherical shells, one inside the other, 
each moveable and each repelled from the sheath with forces, or 
distribution of force, such that the centre of each is attracted 
towards the centre of the hollow with a force varying directly as 
the distance. These suppositions merely put two of Sellmeier's 
single-atom vibrators into one sheath. 

Imagine now a vast number of these diatomic molecules, equal 
and similar in every respect, to be distributed homogeneously 
through all the ether which we have to consider as containing 
sodium-vapour. In the first place, let the density of the vapour 
be so small that the distance between nearest centres is great in 
comparison with the diameter of each molecule. And in the 
first place also, let us consider light whose wave-length is very 
large in comparison with the distance from centre to centre of 
nearest molecules. Subject to these conditions we have (Sellmeier's 
formula) 



where in, m / denote the ratios of the sums of the masses of one 
and the other of the moveable shells of the diatomic molecules in 
any large volume of ether, to the mass of undisturbed ether filling 
the same volume ; K, K I the periods of vibration of one and the 
other of the two moveable shells of one molecule, on the supposition 
that the sheath is held fixed ; v e the velocity of light in pure 
undisturbed ether; v, the velocity of light of period r in the 
sodium- vapour. 

This replaces the concluding portion of the Lecture us originally delivered. It 
was read before the Koyal Society of Edinburgh on Feb. 6, 1899, and reprinted in 
Phil. Mag. for March, 1899, under the title " Application of Sellmeier's Dynamical 
Theory to the Dark Lines Z>,, D 2 produced by Sodium-Vapour." 



DYNAMICS OF FRAUNHOFER*S DARK LINES D^. 177 

For sodium-vapour, according to the measurements of Rowland Molecular. 
and Bell*, published in 1887 and 1888 (probably the most 
accurate hitherto made), the periods of light corresponding to the 
exceedingly fine dark lines D lt D, of the solar spectrum are 
589618 and '589022 of a michronf. The mean of these is so 
nearly one thousand times their difference that we may take 



Hence if we put 

(3); 



and if a: be any numeric not exceeding 4 or 5 or 10, we have 



whence 

T* . 1000 T* 1000 






Using this in (1), and denoting by /* the refractive index from 
ether to an ideal sodium-vapour with only the two disturbing 
atoms m, m t , we find 

1000m . 1000m 



- =u* = 

vj 2*+l 2*-l 

When the period, and the corresponding value of x according 
to (3), is such as to make /** negative, the light cannot enter the 
sodium -vapour. When the period is such as to make /i s positive, 
the proportion, according to Fresnel and according to the most 
probable dynamics, of normally incident light which enters the 
vapour would be 

..(7), 



if the transition from space, where the propagational velocity is 
v e> to medium in which it is v t , were infinitely sudden. 

Judging from the approximate equality in intensity of the 

* Rowland, Phil. Mag. 1887, first half-year; Bell, Phil. Slag. 1888, first half- 
year. 

t See footnote, p. 150. 

T. L. 12 



178 LECTURE XIII. 

Molecular, bright lines D lt A of incandescent sodium- vapour; and from the 
approximately equal strengths of the very fine dark lines D x , _D 2 
of the solar spectrum; and from the approximately equal strengths, 
or equal breadths, of the dark lines D lt D 2 observed in the 
analysis of the light of an incandescent metal, or of the electric 
arc, seen through sodium-vapour of insufficient density to give 
much broadening of either line ; we see that m and ra, cannot be 
very different, and we have as yet no experimental knowledge to 
show that either is greater than the other. I have therefore 
assumed them equal in the calculations and numerical illustrations 
described below. 

At the beginning of the present year I had the great pleasure 
to receive from Professor Henri Becquerel, enclosed with a letter 
of date Dec. 31, 1898, two photographs of anomalous dispersion 
by prisms of sodium-vapour*, by which I was astonished and 
delighted to see not merely a beautiful and perfect demonstration 
of the "anomalous dispersion" towards infinity on each side of the 
zero of refractivity, but also an illustration of the characteristic 
nullity of absorption and finite breadth of dark lines, originally 
shown in Sellmeier's formulaf of 1872 and now, after 27 years, 
first actually seen. Each photograph showed dark spaces on 
the high sides of the D lf D. 2 lines, very narrow on one of the 
photographs ; on the other much broader, and the one beside 
the Do line decidedly broader than the one beside the D l line : 
just as it should be according to Sellmeier's formula, according 
to which also the density of the vapour in the prism must have 
been greater in the latter case than in the former. Guessing 
from the ratio of the breadths of the dark bands to the space 
between their D lt D 2 borders, and from a slightly greater breadth 
of the one beside D.,, I judged that m must in this case have been 
not very different from '0002 ; and I calculated accordingly from 
(G) the accompanying graphical representation showing the value 

of 1 , represented by y in fig. 1. Fig. 2 represents similarly 

the value of 1 for w = '001, or density of vapour five times 

* A description of Professor Becquerel's experiments and results will be found 
in Comptes Rendus, Dec. 5, 1898, and Jan. 16, 1899. 

t Sellmeier, Pogg. Ann. Vol. CXLV. (1872) pp. 399, 520: Vol. CXLVII. (1872) 
pp. 387, 525. 



DYNAMICS OF FRAUNHOFER'S DARK LINES D 
Fig. 1*. TO = -0002. 



179 



Molecnlai 



Fig. 2. 



= -001. 




* In figs. 1 and 2 the D, , 7> 2 lines are touched by carves of finite curvature at 
y = + 1 ; and in figs. 3, 4, and 5 at y = 0. The left-hand side of each dark band is 
an asymptote to the curves of figs. 1 and 2, and a tangent at y = to the curves of 
figs. 3, 4, and 5. The diagrams could not show these characteristics clearly unless 
on a much larger scale. 

122 



180 



LECTURE XIII. 



. Molecular, that in the case represented by fig. 1. Figs. 3 and 4 represent 
the ratio of intensities of transmitted to normally incident light 



Fig. 3. 



-0002. 




i 



999 D, 1000 f\ 1001 1002 

Fig. 4. w = -001. 




Fig. 5. 



MI = -003. 




for the densities corresponding to figs. 1 and 2, arid fig. 5 repre- 
sents the ratio for the density corresponding to the value m = '003. 
The following table gives the breadths of the dark bands for 
densities of vapour corresponding to values of m from '0002 to 
fifteen times that value; and fig. 6 represents graphically the 
breadths of the dark bands and their positions relatively to the 
bright lines D lt D 2 for the first five values of m in the table. 





Breadths of Bands 




-Di 


D 


0002 


09 


11 


0006 


217 


383 


0010 


293 


707 


0014 


340 


1-060 


0018 


371 


1-429 


0022 


392 


1-808 


0026 


408 


2-192 


0030 


419 


2-581 



DYNAMICS OF FRAUNHOFER's DARK LINES AAi- 181 

According to Sellmeier's formula the light transmitted through Molecular 
a layer of sodium-vapour (or any transparent substance to which 

Fig. 6. 




the formula is applicable) is the same whatever be the thickness 
of the layer (provided of course that the thickness is many times 
the wave-length). Thus the Z),, ZX, lines of the spectrum of solar 
light, which has traversed from the source a hundred kilometres 
of sodium-vapour in the sun's atmosphere, must be identical 
in breadth with those seen in a laboratory experiment in the 
spectrum of light transmitted through half a centimetre or a 
few centimetres of sodium- vapour, of the same density as the 
densest part of the sodium-vapour in the portion of the solar 
atmosphere traversed by the light analysed in any particular 
observation. The question of temperature cannot occur except 
in so far as the density of the vapour, and the clustering in groups 
of atoms or non-clustering (mist or vapour of sodium), are con- 
cerned. 

A grand inference from the experimental foundation of Stokes' 
and Kirchhoff's original idea is that the periods of molecular 
vibration are the same to an exceedingly minute degree of 
accuracy through the great differences of range of vibration pre- 
sented in the radiant molecules of an electric spark, electric arc, 
or flame, and in the molecules of a comparatively cool vapour 
or gas giving dark lines in the spectrum of light transmitted 
through it. 

It is much to be desired that laboratory experiments be made, 
notwithstanding their extreme difficulty, to determine the density 
and pressure of sodium-vapour through a wide range of temperature, 
and the relation between density, pressure, and temperature of 
gaseous sodium. 



182 



LECTURE XIII. 



Molecular. Passing from the particular case of sodium, I add an applica- 
tion of Sellmeier's formula, (1) above, to the case of a gas or 
vapour having in its constitution only a single molecular period K. 
Taking m / = in (1), we see that the square of the refractive 
index for values of r very large in comparison with K is 1 + m. 
And remembering that the dark line or band extends through 
the range of values for which (v e /v s y is negative, and that (v e /v g )* 
is zero at the higher border, we see from (1) that the dark band 
extends through the range from 



K to T=T. 



K 

Vl +ra 



.(8). 



As an example suitable to illustrate the broadening of the 
dark line by increased density of the gas, I take m = a x 10~ 4 , 
and take a some moderate numeric not greater than 10 or 20. 
This gives for the range of the dark band from 

T = K to r = /c(l -\d x 10- 4 ) (9); 

and for large values of r it makes the refractive index 
l+^axlO~ 4 , and therefore the refractivity, ^a x 10~ 4 . If for 
example we take a = 6, the refractivity would be '0003, which is 
nearly the same as the refractivity of common air at ordinary 
atmospheric density. 

Fig. 7. 



Asymptote for each curve 




V 

Asymptote, y = 1-0002 




998 



999 



1000 



Asymptote y = 1 



1001 



1002 



DYNAMICS OF A SINGLE DARK LINE. 183 

Taking * = 1000, we have, for values of r not differing from Molecular. 
1000 by more than 10 or 20, 



L-^^t^L, where x = r-100Q (10). 

Thus we have 

/* = 



In fig. 7 the curve marked p. represents the values of the refractive 
index corresponding to values of r through a small range above 
and below K, taking a = 4. The other curve represents the 
proportionate intensity of the light entering the vapour, calculated 
from these values of p by (7) above. 

The table on next page shows calculated values for the ordi- 
nates of the two curves ; also values (essentially negative) for the 
formula of intensity calculated from the negative values of /* 
algebraically admissible from (11). 

The negative values of ft have no physical interpretation for 
either curve ; but the consideration of the algebraic prolongations 
of the curves through the zero of ordinates on the left-hand side 
of the dark band illustrates the character of their contacts. The 
physically interpreted part of each curve ends abruptly at this 
zero ; which for each curve corresponds to a maximum value of x. 
The algebraic prolongation of the /* curve on the negative side is 
equal and similar to the curve shown on the positive side. But 
the algebraic prolongation of the intensity curve through its zero, 
as shown in the table, differs enormously from the curve shown on 
the positive side. To the degree of approximation to which we 
have gone, the portions of the intensity curve on the left and 
right hand sides of the dark band are essentially equal and 
similar. This proves that so far as Sellmeier's theory represents 
the facts, the penumbras are equal and similar on the two sides 
of a single dark line of the spectrum uninfluenced by others. It 
is also interesting to remark that according to Sellmeier as now 
interpreted, the broadening of a single undisturbed dark line, 
produced by increased density of the gas or vapour, is essentially 
on the high side of the finest dark line shown with the least 
density, and is in simple proportion to the density of the gas.] 



184 



LECTURE XIII. 



Molecular. 



X 


M 


i ^~ 1 V 
"U+u 


/a positive 


H negative 


-1-0 


894 






-9 


882 






-8 


866 






-7 


845 


993 


-141-9 


-6 


816 


988 


- 82-3 


-5 


774 


984 


-61-5 


-4 


707 


971 


- 33-5 


-3 


"577 


928 


-12-9 


-2 











-1 


imaginary 


imaginary 


imaginary 





00 








+ 1 


1-732 


928 


-12-9 


+ '2 


1-414 


971 


- 33-5 


+ 3 


1-291 


984 


-61-5 


+ '4 


1-247 


988 


-82-3 


+ '5 


1-183 


993 


-141-9 


+ 6 


1-155 






+ '7 


1-134 






+ '8 


1-118 






+ 9 


1-105 






+ 1-0 


1-095 







LECTURE XIV. 
MONDAY, October 13, 8 P.M. 

AT this lecture were seen, immediately behind the model Molecular, 
heretofore presented, two wires extending from the ceiling and 




bearing a long heavy bar about three feet above the floor by 
means of closely-fitting rings. By slipping these rings along the 
bar, the period of vibration of this bifilar suspension could be 
altered at will. Two parallel pieces of wood, jointed at each end, 



186 LECTURE XIV. 

Molecular, served to transmit the azimuthal motion of this vibrator to the 
lower bar P of the model. 

Let us look at this and see what it does. I have not seen it 
before, and it is quite new to me. Oh, see, you can vary the 
period; that is very nice, that is beautiful. We are going to 
study these vibrations a little, just as illustrations. Prof. Rowland 
has kindly made this arrangement for us, and I think we will all 
be interested in experimenting with it. We have this bar P, 
moved by this bifilar pendulum H, which is so massive that its 
period is but little affected, I suppose, by being connected with P. 
It takes some time before the initial free vibrations in the model 
are got quit of and the thing settles into simple harmonic motion 
corresponding to the period of the bifilar pendulum. If we keep 
this pendulum going long enough through nearly a constant range, 
the masses P, m 1} m 2> m 3 will settle into a definite simple har- 
monic motion, through the subsidence of any free vibrations which 
may have been given to them in the start. We see the whole 
apparatus now performing very nearly a simple harmonic motion. 
We will now superimpose another vibration on this by altering 
the period of the pendulum very slightly. That, you see, seems 
to have diminished very much the vibrations of the system. They 
are now increasing again. That will go on for a long time. I 
shall give this pendulum a slight impulse when I see it flagging, 
to keep its range constant. When it is in its middle position, 
I apply a working couple. We will give no more attention to it 
than just to keep it vibrating, while we look at these notes which 
you have in your hands, and which I have prepared for you so 
as to shorten our work on the black-board. 

[These notes related to the tasinomic treatment of asolotropy. 
The discussion of them was interrupted at intervals to continue 
experiments and observations on Professor Rowland's model. That 
part of the Lecture has been omitted from the print as the subject 
has been treated in the addition of date February 14, 1899 to 
Lecture XIII., pp. 170175.] 

Let us stop and look at our vibrating apparatus. It has been 
going a considerable time with the exciter kept vibrating through 
a constant range, and you see but small motion transmitted to 
the system. That is an illustration of the most general solution 
of our old problem*. Our " handle " P is in firm connection with 
* Lecture III. Pt. 2, p. 38. 



ROWLAND'S MODEL VIBRATOR. 187 

the bifilar pendulum (the exciter) and is forced to agree with it. Molecular. 
Let us bring the system to rest. Now start the exciter and keep 
it going. In time the viscosity will annul the system of vibrations, 
representing the difference between zero and the permanent state 
of vibration which the three particles will acquire. If there were 
no loss of energy whatever, the result would be that this initial 
jangle, which you now see, would last for ever; consisting of 
a simple harmonic motion in the exciter and a compound of the 
three fundamental modes of these three particles viewed as a 
vibrating system with the bar P held fixed. Let the system with 
the bar P held fixed be set to vibrate in any way whatever; then its 
motion will be merely a compound of those three fundamental 
modes. But now set the exciter going, and the state of the case 
may be looked on as thus constituted ; the exciter and the whole 
system in simple harmonic motion of the same period, and, superim- 
posed upon that, a compound of the three modes of simple harmonic 
vibration that the system can perform with the exciter fixed. We 
cannot improve on the mathematical treatment by observation; 
and really a model of this kind is rather a help or corrective to brain 
sluggishness than a means of observation or discovery. In point 
of fact, we can discover a great deal better by algebra. But 
brains are very poor after all, and this model is of some slight 
use in the way of making plain the meaning of the mathematics 
we have been working out. 

The system seems to have come once more into its permanent 
state. Let us stop the exciter and see how long the system will 
hold its vibration. The reaction on the exciter is very slight, it is 
very nearly the same as if that massive bar H were absolutely 
fixed. But the motion actually communicated to it, since it is 
not absolutely fixed, will correspond to a considerable loss of 
energy. A very slight motion of H with its great length and 
mass has considerable energy compared even with the energy of 
our particle of greatest mass ; so that our system will come to 
rest far sooner than if H were absolutely fixed. The model is at 
present illustrating phosphorescence. You see the particles (nt,, 
?,, m,} have gone on vibrating for a whole minute, and w, must 
have performed a couple of dozen vibrations at least. A true 
phosphorescence of a hundred seconds' duration is quite analogous 
to the giving back of vibrations which you see in our model, 
only instead of our two or three dozen vibrations, we have in 



188 LECTURE XIV. 

Molecular, phosphorescence 40,000 million-million vibrations during a hundred 
seconds. Now, we cannot get 1000 residual vibrations in our 
model because of the dissipation of energy arising from imperfect 
elasticity in the wire, friction between parts of the model, and the 
resistance of the air. That dissipation of energy is simply the 
conversion of energy from one state of motion, (the visible motions 
which we have been watching), into another (heat in the wire, 
heat of frictions, and heat in the air). In molecular dynamics, we 
have no underground way of getting quit of energy or carrying it 
off. We must know exactly what has been done with it when the 
vibration of an embedded molecule ends, even though this be 
not before a thousand million-million periods have been performed. 
[March 6, 1899. Imagine a homogeneous mass of rock granite or 
basalt, for example as large as the earth, or as many times larger as 
you please, but with no mutual gravitation between its parts to 
disturb it. Let there be, anywhere in it very far from a boundary, 
a spherical hollow of 5 cms. radius, and let a violin-string be 
stretched between two hooks fixed at opposite ends of a diameter 
of this hollow, and tuned to vibrations at the rate of 1000 per 
second. Let this string be set in vibration (for the present, no 
matter how) according to its gravest fundamental mode, through a 
range of one millimetre. Let the elasticity of the string and of the 
granite be absolutely perfect, and let there be no air in the hollow 
to resist the vibrations. They will not last for ever. Why not ? 
Because two trains of waves, respectively condensational-rarefac- 
tional and purely distortional, will be caused to travel outwards, 
carrying away with them the energy given first to the vibrating 
string (see below 28 of addition to Lecture XIV.).] We must 
suppose the elasticity of our matter and molecules to be perfect, and 
we cannot in any part of our molecular dynamics admit unaccounted 
for loss of energy ; that is to say, we cannot admit viscous terms 
unless as an integral result of vibrations connected with a part of 
the system that is not convenient for us to look at. 

In three minutes our system has come very nearly to rest. 
We infer therefore that in three minutes from a commencement of 
vibration of the exciter we shall have nearly reached the permanent 
state of things. 

Now wo vary the period of the exciter, making it as nearly as 
we can midway between two fundamental periods of our complex 
molecule. We will keep this going in an approximately constant 



MOTION OF ETHER WITH EMBEDDED MOLECULES. 189 

range for a while and look at the vibrations which it produces in Molecular, 
the system. 

Now you see very markedly the difference in the vibrations 
of our system after it has been going for several minutes with 
the exciter in a somewhat shorter period of vibration than that 
with which we commenced. Here is another still shorter. In 
the course of two or three minutes the superimposed vibrations 
will die out. See now the tremendous difference of this case in 
which the period of the exciter is approximately equal to one 
of the fundamental periods of the system, or the periods for the 
case in which the lowest bar is held absolutely fixed. 

I had almost hoped that I would see some way of explaining 
double refraction by this system of molecules, but it seems more 
and more difficult. I will take you into my confidence to-morrow, 
if you like, and show you the difficulties that weigh so much upon 
me. I am not altogether disheartened by this, because of the 
fact that such grand and complicated and highly interesting 
subjects as I have named so often, absorption, dispersion and 
anomalous refraction, are all not merely explained by their 
means but are the inevitable results of this idea of attached 
molecules. 

There is one thing I want to say before we separate, and that 
is, when I was speaking last of the subject, I saw what seemed 
to me to be a difficulty, but on further consideration, I find it 
no difficulty at all. Not very many hours after I told you it 
was a difficulty, I saw that I was wrong in making it appear 
to be a difficulty at all. I do not want to paint the thing any 
blacker than it really is and I want to tell you that that question 
I put as to the ether keeping straight with the molecules is 
easily answered when there is a large number. Our assumption 
was a large number of spherical cavities, lined with rigid spherical 
shells and masses inside joined by springs or what not : and the 
distance from cavity to cavity small in comparison with the wave- 
length. It then happens that the motion of the medium rela- 
tively to the rigid shells will be exceedingly small and a portion 
of the medium that will contain a large number of these shells will 
all move together (see below, addition to Lee. XIV.). If the distance 
from molecule to molecule is very small in comparison with the 
wave-length, then you may look upon the thing as if the structure 
were infinitely fine, and you may take it that the ether moves 



190 LBC-mtE 5JV. 

:: ~e ^T7-L : r-- wi-h. the iZ. and no* in and oat among them. 
as I sai'i IT is evident :. 'he och-er hand, when the wave- 
ibe light sarersng the medium is moderate in 
with the distance between the molecules, thai it 
:~ss move oci and in among them. Bat if the srrffness of tfce 
meif^m is s^ch is ire- rrake nfe wave-length Lar^-e in comparison 
viih -be .ii^aji'^r r:-ni m.xecale to mc-iectile- this >rifm^y is a- 
i:i-fn~ T.: keep -feem all n:>ge5her. and yoo. may regard these ris^d 
sbeLlf is c:c.i? -:f arachment by which the m:4eeale is polled 
-'- : ~ -F-iT anI "has way. : that we niay suppose oar reaeti-ocHLnr 
5:r:-ef. :: whieh c-ig x-^is a sample, to be absolutely the same 
n. :h-ir eeo' npi^n the medium as if they were uniformlr 
iLf^rf'^-eii -hroogfa. ii. 

Tha: ^ikes away :ce port of our disor-nient. The only dini- 
?r^Ty Tbiu I see just now is that of explaining doable refraction. 
The s-bject grc'ws up:-n us terribly, and so does our want of 
-- -. If i: is not too much for you I must have one of oar 
docibLe lectures tc-morrow. 

[Twice* in this Lecture, and indeed many times in preceding 
and subsequent Lectures. I felt the want of a full mathematical 
inTes-i^idon of spherical waves originating in the application of 
r';-rce :..- an elastic solid within a limited space. I have therefore 
recently undertaken this work^. and I a^ve the following 1 statement 
of :: as an addition to Lecture XIV. 

1. The complete mathematical theory of the propagation of 
motion through an infinite elastic solid, including the analysis of 
the motion into two species, equivoluminal and irrotational. was 
first given by Stokes in his splendid paper "On the Dynamical 
Theory of Diffraction*." The object of the present communication 
i* to investigate fully the forcive which must be applied to the 
boundary, 5. of a hollow of any shape in the solid, in order to 
originate and to maintain any known motion of the surrounding 
solid: and to solve the inverse problem of finding the motion 



* p. L r and pp. 1^9, 190. 

t Comirnnvrated to E. S. E. on May 1, 1899. and published in Pkil. 3fa</. May. 
Aoe. and Oct., l-'.)9 onder the title w On the Application of Force within a Limited 
Spa<ee. rwoired to produce Spherical Solitary Waves, or Trains of Periodic Wares, 
of both species. Eqni^olnminai and IrrotationaL in an Elastic Solid." 

~ Stokes, Mathematical Pa-pen, YoL n. p. 243. 



SPHERICAL WAVES IN ELASTIC SOLID, 



101 



when the forcive on, or the motion of, S is given, for the particular Molr. 
case in which S is a spherical surface kept rigid. 

2. Let , 17, denote the infinitesimal displacement at any 
point of the solid, of which (x, y, z) is the equilibrium }KMiition. 
The well-known equations of motion* are 



where B denotes - + -~ + -^ . Using tho notution of ThoniNoii 
and Taitf for strain-components (elongation! ; and distortions), 
*. /. 9 i 6, c ; we have 



*>. 

dy' 



dr, d 

-J- -f , ; 
dz dy 



dy 



(2); 



and with the corresponding notation P, Q, It ; H, T, U, for troi- 
components (normal and tangential forc- on th xix hi<l'-H of un 
infinitely small rectangular parallelepi^d), w- huvc 



S=na; T=nb; U-nc 

Let now o- be an infinitesimal area at any (x/int of th<: 
S; X, /i, y the direction-cosines of the normal ; and A'//, >'//. /'/ 
the components of the force which must be applied from within U/ 
produce or maintain the specified motion of the 
We have 




< M -,- 



Ml 



192 LECTURE XIV. 

Molar. whence by (3) 



vb) \ 

\c)\ ......... (5). 

- Z = (k - f n) vB + n (2vg + \b + pa)} 

These equations give an explicit answer to the question, What 
is the forcive ? when the strain of the matter in contact with S is 
given. We shall consider in detail their application to the case in 
which S is spherical, and the motions and forces are in meridional 
planes through OX and symmetrical round this line. Without 
loss of generality we may take 

z = Q\ giving v = Q, a = 0, 6 = 0, Z = ............ (6). 

Equations (5) therefore become 

-X = (k- f n) \B + n (2\e + /*c) 7 



3. In 5 26 of his paper already referred to, Stokes gives 
a complete solution of the problem of finding the displacement 
and velocity at any point of an infinite solid, which must follow 
from any arbitrarily given displacement and velocity at any 
previous time, if after that, the solid is left to itself with no 
force applied to any part of it. In a future communication I 
hope to apply this solution to the diffraction of solitary waves, 
plane or spherical. Meantime I confine myself to the subject 
stated in the title of the present communication, regarding which 
Stokes gives some important indications in 27 29 of his paper. 

4. Poisson in 1819 gave a complete solution of the equation 

-* .............................. (8) 

in terms of arbitrary functions of x, y, z, representing the initial 
values of w and -j- ; and showed that for every case in which w 
depends only on distance (r) from a fixed point, it takes the form 



where F and / denote arbitrary functions. In my Baltimore 
Lectures of 1884 (pp. 46, 86, 87 above) I pointed out that solutions 



SPHERICAL WAVES IN ELASTIC SOLID. 193 

expressing spherical waves, whether equivoluminal (in which there Molar 
is essentially different range of displacement in different parts of 
the spherical surface) or irrotational (for which the displacement 
may or may not be different in different parts of the spherical 
surface), can be very conveniently derived from (9) by differen- 
tiations with respect to x, y, z. It may indeed be proved, although 
I do not know that a formal proof has been anywhere published, that 
an absolutely general solution of (8) is expressed by the formula 



r = J[(x - xj + (y- yj + (z - zj] ........... (10), 

where S denotes sums for different integral values of h, i, j, and 
for any different values of x', y', z'. 

5. I propose at present to consider only the simplest of all 
the cases in which motion at every point (x, 0, 0) and (0, y, z) is 
parallel to X'X ; and for all values of y and z, is the same for 
equal positive and negative values of x. For this purpose we of 
course take x' = y' = z = ; and we shall find that no values of h, 
i,j greater than 2 can appear in our expressions for f, rj, , because 
we confine ourselves to the simplest case fulfilling the specified 
conditions. Our special subject, under the title of this paper, 
excludes waves travelling inwards from distant sources, and there- 
fore annuls f(t + rfv). 

6. In 5 8 of his paper Stokes showed that any motion 
whatever of a homogeneous elastic solid may, throughout every 
part of it experiencing no applied force, be analysed into two 
constituents, each capable of existing without the other, in one of 
which the displacement is equivoluminal, and in the other it is 
irrotational. Hence if we denote by ((,, 17,, ,) the equivoluminal 
constituent, and by (,, 17, , a) the irrotational constituent, the 
complete solution of (1) may be written as follows: 



where ,, 77,, & and ,, ?/ 2 , > fulfil the following conditions, (12) 
and (13), respectively : 



T.L. 13 



194 LECTURE XIV. 

Molar. fc _ dw . _ d W . .. 

^~~dx> ^"""dy 1 t2 
w being any solution of 



.(13). 



The first equation of (12) shows that in the ( 1; 17,, ) con- 
stituent of the solution there is essentially no dilatation or con- 
densation in any part of the solid ; that is to say, the displacement 
is equivoluminal. The first three equations of (13) prove that in 
the (f 2 , ?;,, 2) constituent the displacement is essentially irrota- 
tional. 

7. We can now sec that the most general irrotational 
solution fulfilling the conditions of 5 is 

t _ d "~ F JI d ' ?*. r - ^ ?* 

^~di? r ' ^ ~ d.fdy r ' **~ dxdz r " 
giving 

f + ** + tf! = \**! .................. ( H'); 

ax ay dz v~ dx r 

and the most general equivoluminal solution fulfilling the same 
conditions is 



giving 

'- -j- -| = (15'), 

dx dy dz 

where F^ and F 2 are put for brevity to denote arbitrary functions 

of (t j and It I respectively. Hence the most general solu- 
\ u/ \ v j 

tion fulfilling the conditions of 5 is 

dx 2 r u-r ' dxdy r ' J-J--" 1 



where for brevity < denotes a function of r and t, specified as 
follows : 



(17). 
Denoting now by accents differential coefficients with respect to r, 



SPHERICAL WAVES IN ELASTIC SOLID. 



195 



and retaining the Newtonian notation of dots to signify differential Molar, 
coefficients with reference to t, we have 



Working out now the differentiations in (1C), we find 

^"^"wV 

8. For the determination of the force-components by (7), we 
shall want values of 8, e.f, and c. Using therefore (2) and going 
back to (16) we see that 

e.j*!'**... ...(20). 

ax M* ay r 

Hence, and by (19), we find 

By (16), (14'), and (15') we find 

and by (19) directly used in (2) we find 

"r*" + ~r*~ " r* ) + 1* ~ V + r , 

(23). 

Remark here how by the summation of these three formulas we 
find fur e +f+ ff the value given for 8 in (22). 

132 



196 



LECTURE XIV. 



9. These formulas (22) and (23), used in (5), give the force- 
components per unit area at any point of the boundary (S of 1) 
of a hollow of any shape in the solid, in order that the motion 
throughout the solid around it may be that expressed by (19). 
Supposing the hollow to be spherical, as proposed in 2, let its 
radius be q. We must in (23) and (5) put 



(24); 



and putting v = 0, we have, as in (7), the two force-components for 
any point of the surface in the meridian z = 0, expressed as 
follows : 



- n [2\*A + 2 (2\ 2 + 1) B + (\= 
Y = (k - - w ) C a \fji - nV (2A + 4B + C. 2 ) 
where 



<$>'" G ( J>" 154>' 

A = + 

q q 2 q s q* 



n _^_ _ 3W W 

~ q- q* q* 



(26), 



qu 3 qti 

a = ^ + -|: 

qv } q 2 v 2 
<I> and $ denoting (f> and F with q for r. 

10. Returning now to (19), consider the character of the 
motion represented by the formulas. For brevity we shall call 
XX' simply the axis, and the plane of YY', ZZ' the equatorial 
plane. First take y = 0, z = 0, and therefore x = r. We find by 
aid of (18) 



Next take as = and we find 
(equatorial) f = - -i _ 



.(28). 



SPHERICAL WAVES IN ELASTIC SOLID. 



197 



Hence for very small values of r we have 
(axial) =-?( 



Molar. 



.(29); 



.(30). 



(equatorial) % = - - (F, + F 2 ) 
and for very large values of 7*, 
(axial) = ? 

(equatorial) % = 

Thus we see that for very distant places, the motion in the 
axis is approximately that due to the irrotational wave alone ; and 
the motion at the equatorial plane is that due to the equivolnminal 
wave alone : also that with equal values of F t and F. t the equi- 
voluminal and the irrotational constituents contribute to these 
displacements inversely as the squares of the propagational 
velocities of the two waves On the other hand, for places very 
near the centre, (29) shows that both in the axis and in the 
equatorial plane the irrotational and the equivoluminal con- 
stituents contribute equally to the displacements. 

11. Equations (25) and (26) give us full specification of 
the foreive which must be applied to the boundary of our spherical 
hollow to cause the motion to be precisely through all time that 
specified by (19), with F l and F. 2 any arbitrary functions. Thus 
we may suppose F t (t q/u) and F. 2 (t q/v) to be each zero for all 
negative values of t, and to be zero again for all values of t exceed- 
ing a certain limit T. At any distance r from the centre, the 
disturbance will last during the time 



from 



and from 



to 



to t = 



_r-q 



.(31). 



u u 

Supposing v > u, we see that these two durations overlap by 
an interval equal to 



if 



r - q < r 

1 



T- 1 ) 

/ \u vJ 



198 



LECTURE XIV. 



On the other hand, at every point of space outside the radius 
q+r/(l/u lfv) the wave of the greater propagational velocity 
passes away outwards before the wave of the smaller velocity 
reaches it, and the transit-time of each wave across it is r. The 
solid is rigorously undisplaced and at rest throughout all the spaces 
outside the more rapid wave, between the two waves, and inside 
the less rapid of the two. 

12. The expressions (25) and (26) for the components of the 
surface-forcive on the boundary of the hollow required to produce 
the supposed motion, involve J/^ and J^ 2 - Hence we should have 
infinite values for t = or t = r, unless F, and F z vanish for t = 
and t = r, when r = <j. Subject to this condition the simplest 
possible expression for each arbitrary function to represent the two 
solitary waves of 11, is of the form 

^ r= ( 1 -% 2 ) 3 > where X = --l (33). 

Hence, by successive differentiations, with reference to t, 



(34). 



The annexed diagram of four curves represents these four 
functions (33) and (34). 




13. Take now definitively 



.(35), 



where 



*-('- 



.(36). 



SPHERICAL WAVES IN ELASTIC SOLID. 



199 




Molar. 



Scale 



of *= scale of F. 



200 



LECTURE XIV. 



Molar. Consider now separately the equivoluminal and the irrotational 
motions. Using (19), (18), (35), (34), and taking the equi- 
voluminal constituents, we have as follows: 



Equatorial, x = ; rj l = 0, 



(37), 



Cone of latitude 45, x- xy 



.(38), 



Axial, 2 = 



(39); 



where 



14. Similarly for the irrotational constituents; 
[Equatorial, x = ; tj., = 0, 
' 3 1 2(j s 

7 ;,/' , J --y ,/>'/ 



Cone of latitude 45, a" = xy = \r z , 



(43); 



where ^ &c. are given by (40) with %., for ^j ; ^ and ^ 2 being 
given by (36). 

15. The character of the motion throughout the solid, which 
is fully specified by (19), will be perfectly understood after a 
careful study of the details for the equatorial, conal, and axial 



SPHERICAL WAVES IN ELASTIC SOLID. 201 

places, shown clearly by (37)... (43) for each constituent, the equi- Molar, 
voluminal and the irrotational, separately. The curve 3 in the 
diagram of 12 shows the history of the motion that must be given 
to any point of the surface S, for either constituent alone, and 
therefore for the two together, in any case in which q is exceed- 
ingly small in comparison with the smaller of the two quantities 
UT, vr, which for brevity we shall call the wave-lengths. The curve 
Jfr shows the history of the motion produced by either wave when 
it is passing any point at a distance from the centre very great in 
comparison with its own wave-length. But the three algebraic 
functions 3, 3, $ all enter into the expression of the motion 
due to either wave when the faster has advanced so far that its 
rear is clear of the front of the slower, but not so far as to make 
its wave-length (which is the constant thickness of the spherical 
shell containing it) great in comparison with its inner radius. 
Look at the diagram, and notice that in the origin at S, a mere 
motion of each point in one direction and back, represented by J?, 
causes in very distant places a motion (,) to a certain displace- 
ment d, back through the zero to a displacement 1 36 x d in the 
opposite direction, thence back through zero to d in the first 
direction and thence back to rest at zero. Remark that the 
direction of d is radial in the irrotational wave and perpendicular 
to the radius in the equivoluminal wave. Remark also that the 
d for every radial line varies inversely as distance from the 
centre. 

16. Draw any line OPK in any fixed direction through 
0, the centre of the spherical surface S at which the forcive 
originating the whole motion is applied. In the particular 

case of 12... 15, and in any case in which F l it % j and 

F, It - ) are each assumed to be, from t = to t = r, of the 

form t?(T tyAit\ where i denotes an integer, the time-history 
of the motion of P is B 9 + Bj + . . . -I- B 6+i P +i , and its space-history 
(t constant and r variable) is CL,7-- S + C_r- + ... + C t+i f* +i ; the 
complete formula in terms of t and r being given explicitly by (19). 
The elementary algebraic character of the formula : and the exact 
nullity of the displacement for every point of the solid for which 
r > q + vt ; and between r = q + v (t T) and r = q + ut, when 



202 LECTURE XIV. 

Molar. v(t r)> ut ; and between r = q and r = q + u (t r), when t > r ; 
these interesting characteristics of the solution of a somewhat 
intricate dynamical problem are secured by the particular character 
of the originating forcive at S, which we find according to 8, 9 
to be that which will produce them. But all these characteristics 
are lost except the first (nullity of motion through all space out- 
side the spherical surface r = q + vt), if we apply an arbitrary 
forcive to S*, or such a forcive as to produce an arbitrary deforma- 
tion or motion of S. Let for example 8 be an ideal rigid spherical 
lining of our cavity; and let any infinitesimal arbitrary motion 
be given to it. We need not at present consider infinitesimal 
rotation of >S f : the spherical waves which this wotdd produce, 
particularly simple in their character, were investigated in my 
Baltimore! Lectures^, and described in recent communications to 
the British Association and Philosophical Magazine^. Neither 
need we consider curvilinear motion of the centre of S, because 
the motion being infinitesimal, independent superposition of ;<-, y-, 
z-motion.s produces any curvilinear motion whatever. 

17. Take then definitively (> (t), or simply , an arbitrary 
function of the time, to denote excursion in the direction OX, of 
the centre of N from its equilibrium-position. Let d" 1 /', d~ s 

ft rt ft 

denote I dff'' and I <lt dtt'. Our problem is, supposing the solid 

J II .' (I . U 

to be everywhere at rest and unstrained when t = 0, to find (, 77, ) 
lor everv point of the solid (r>q) at all subsequent time (t posi- 
tive); with 

at r=q, = (t), rj=(), =0 (44). 

These, used in (It)), give 

o,A() + M) + 3iir^ + ^l + \ W ( 0+ ^(0]} 

u* v- ( V [_ v J <r ) 

(45), 

* If the space inside X is filled with solid of the same quality as outside, the 
solution remains algebraic, if the forcive formula is algebraic, though discontinuous. 
The displacement of .S' ends, not at tinio t-r when the forcive is stopped, but at 
time t = T + '2qlu when the hist of the inward travelling wave produced by it has 
travelled in to the centre, and out again to r q. 

t Pp. HI KB and 159, 100 above. 

J n. A. llrport, 1H)8, p. 783; Phil. Mng. Nov. 181)8, p. 41(4. 



SPHERICAL WAVES IN ELASTIC SOLID. 203 

and Molar. 



Adding % x (46) to (45), we find 

-2^ + J^ 



whence 

ST. /'A ST. m 

(48); 



and by this eliminating ,? a from (46), 

'+2*)]jr I (o=e?(o ...... (40), 

where d denotes d/dt, and 

c*'(0 = -'/"'(i + a-'+yja-V(0 ............... (so). 

18. I hope later to work out this problem for the case of 
motion commencing from rest at t = 0, and & (t) an arbitrary 
function ; but confining ourselves meantime to the case of S 
having been, and being, kept perpetually vibrating to and fro in 
simple harmonic motion, assume 

(Q = /< Hiii orf ........................... (51). 

With this, (50) gives 

^(0-^ f [(~i-l)n^+^corf] ............ (52). 

To solve (40) in the manner most convenient for this form of ri>'(t), 
we now have 



- (tt + 2v) 9 + - f (u 

s --i(u' + 2y s )-- 



204 

Molar. = hqu* X 



LECTURE XIV. 



(u? + 2v 2 ) + - (v- + 3vu u 2 } + ft)4 sin wt H ( + &> 2 ) cos cat 

a*a>* ^ (/" g m ^ ( 1 



.(53). 



With J*i thus determined, (48) gives JX 2 as follows, 



v 2 w 2 &)'- 

For f, 77, f by (19) we now have 



(51). 






wliere, with notation corresponding to (20) above, 



| 19. The wave-lengths of the equivoluminal and rotational 
waves are respectively ' and . For values of r very great 

d) 0) 

in comparison with the greater of these, the second members of 
(55) become reduced approximately to the terms involving f\ and 
F y . These terms represent respectively a train of equivoluminal 
waves, or waves of transverse vibration, and a train of irrotational 
waves, or waves of longitudinal vibration ; and the amplitude of 
each wave as it travels outwards varies inversely as r. 

20. For the case of an incompressible solid we have v = oo , 
which by (53) gives 



wt. 



.(57); 



SPHERICAL WAVES IN ELASTIC SOLID. 205 

and by (55) we have, for r very great, Molar. 



.(58), 



rj = %hq sin tat ^ 

*i * xz 
f = \hq sin tat 

which fully specify, for great distances from the origin, the wave- 
motion produced by a rigid globe of radius q, kept moving to and 
fro according to the formula h sin tat. 

21. The strictly equivoluminal motion thus represented 
consists of outward- travel ling waves, having direction of vibration 
in meridional planes, and very approximately* per]>endicular to 

the radial direction, and amplitude of vibration equal to h -- sin 6, 

where 6 denotes the angle between r and the axis. The gradual 
change from the simple motion =//sino> at the surface of the 
rigid globe, through the elastic solid at distances moderate in 
comparison with q, out to the greater distances where the motion 
is very approximately the pure wave-motion represented by (5H), 
is a very interesting subject for detailed investigation and illustra- 
tion. The formulas expressing it are found by putting v= oo in 
(45), and using this equation to determine F 3 (t) in terms of F t (t)] 
then using (47) to determine F } (t) in terms of & (t) given by (51) ; 
and then using (55) and (5(i), for which v = oo makes &, = t, to 
determine f, tj, . They are as follows: 

3it a \ . 3u 5 . 3u 
1 sin wt H ^~2 Sin W M coscoc 



(59), 



* Rigorously so, if the wave-length and q are each infinitely small in comparison 
with r, 



206 LECTURE XIV. 

Molar. where 

r-B(r, f) = MM V | f 1 ~ T~s) s * n wt + ~iT"2 S ^ n wtl ~ 



t, = t- 




- cos w^ - *- -g- sin o)*,, 

\r/ 2 <<o ?' 2 



....(60). 



ftk-^ 






For the particular case of the wave-length equal to the radius we 
have 



2-7TM 



...(61), 



SPHERICAL WAVES IN ELASTIC SOLID. 



207 



which enables us to write simply l/2?r for u/qa> in equations (59) Molar. 
and (60). For graphic representation of this case we take z = 0, 
which makes all the displacements lie in the plane (xy). 

22. The accompanying drawings help us to understand 
thoroughly the character of the motion of the solid throughout 
the whole infinite space around the vibrating rigid globe. They 
show displacements and motions of points of which the equilibrium 
positions are in the equatorial plane, in the cone of 45 latitude, 







* 




























































' 







































































































































































,' 




















































, 














































































































, 








































.- 


















. 





























































. 






























- 





















































- 


* 



















































.' 




































^ 





























































. 




-. 


. 


































































I 






















































- 


. 









- 












































1 


- 


- 












s 


























































N 


\ 
































- 














" 


M 
.- *- 


"W. -4 

/- 


, > 


-.4. 
-1 1 


-/. 


K^ 


^. 








^ 







































i 


- 




! 






| 






i 























































































. 










































- 




















































.-.. 


.... 


* 




~. . 






.- 

































































. 






. 


- 










































I 





1 
8. 






i 


i 




i 


' 




i 







-.- 






and in the axial line. Fig. 1 represents displacements at an 
instant when the globe is moving rightwards through its middle 
position. Fig. 2 shows displacements a quarter- period later, when 
the globe is at the end of its right ward motion. Each figure shows 



208 LECTURE XIV. 

also the orbit of a single particle of which the equilibrium position 
is in the 45 cone, at a distance f q from the centre of the globe. 
The orbital motion is in the direction of the hands of a watch. It 
is interesting to see illustrated in fig. 2 how the axial motion is 
gradually reduced from + h at the surface of the globe to a very 
small range at distance q from the surface, or 2q from the centre, 
and we are helped to understand its gradual approximation to 
zero at greater and greater distances by the little auxiliary 
diagrams annexed, in which are shown by ordinates the magnitudes 
of the axial displacements at the two chosen times. 

23. The gradual transition from motion h sin att parallel to 
the axis at the surface of the globe, to motion 

- - h sin 6 sin wt 
2 r 

at great distances from the globe in any direction, is interestingly 
illustrated by the conal representations in the two diagrams for 
the case 6 = 45. It should be remarked that in reality h ought 
to be a small fraction of </, the radius of the globe, practically not 
more than T i- rt , in order that the strains may be within the limits 
of elasticity of the most elastic solid, and that the law of simple 
proportionality of stresses to strains (Hooke's Ut tensio sic vis) may 
be approximately true. In the diagram we have taken h = ^q; 
but if we imagine every displacement reduced to ^ of the amount 
shown, and in the direction actually shown, we have a true, highly 
approximate representation of the actual motions, which would be 
so small as to be barely perceptible to the eye, fora globe of 6 cms. 
diameter. 

24. Return now to our solution (53), (54), (55), (56) for 
arbitrary or periodic motion of a rigid globe embedded in an 
isotropic elastic solid of finite resistance to compression and finite 
rigidity. For distances from the globe very great in comparison 
with q, its radius, that is to say for q/r very small, (55) and (5G) 
become 



(62); 
tj = B (r, t)xy; % = B (r, t) xz 



SPHERICAL WAVES IN ELASTIC SOLID. 



209 






where 






Putting now in the equations (62) the value of B (r, t) from 
(63), and eliminating $M) by (47), we find 



xy**i(t t ) , ' r >/\ 9 **i(t* 

*>=> 75 -, + r u 



...(6*). 



The terms of these formulas, having , and /., respectively for 
their arguments, represent two distinct systems of wave-motion, 
the first equivoluminal, the second irrotational, travelling outwards 
from the centre of disturbance with velocities u and v. 

2"). I reserve for some future occasion the treatment of the 
case in which (t) is discontinuous, beginning with zero when 
t = and ending with zero when t = r. \ only remark at present 
in anticipation that $i(t\ determined by the differential equation 
(49), though commencing with zero at t = 0, does not come to zero 
at t = r, but subsides to zero according to the logarithmic law 
(e~ w ) as t goes on to infinity; and that therefore, as the same 
statement is proved for ^? s (0 by (4S), neither the equivoluminal 
nor the irrotational wave-motion is a limited solitary wave of 
duration r, but on the contrary each has an infinitely long 
subsidential rear. 

26. For the general problem of the globe kept in simple 
harmonic motion, h sin ait, parallel to OX, we may write (53) 
for brevity as follows : 



T. L. 



14 



210 



where 



LECTURE XIV. 

J/TJ (A -JL- (K sin a>t+L cos <*>), 
ft)" 

' 4 V 4 (X + 2 ^) + r 2 2 ('^ + % VU ~ "'0 + A 



(u 2 + 2tf-) - 1 " + , 
[tfu* q-a- 



u v f& 



1 = 



1 " 9 ) ON |- 1 T 9>V 

In terms of this notation, (6-i) gives for great distances from the 
centre 



These equations represent two sets of simple harmonic waves, 
equivoluminal and irrotational, for which the wave-lengths are 
respectively 27TM/&>, 2?n;/&). The maximum displacements in the 
two sets at points of the cone of semi- vertical angle 6 and axis OX, 
are respectively, 



(equivoluminal) sin 6 ~ 



(irrotational) cos 6 ^ V[(3 - 2/0 2 + 47,-] 



(67). 



27. The rate of transmission of energy outwards by a 
single set of waves of either species is equal, per period, to the 
sum of the kinetic and potential energies, or, which is the same, 
twice the whole kinetic energy, of the medium between two 
concentric spherical surfaces, of radii differing by a wave-length. 
Now the average kinetic energy throughout the wave-length in 
any part of the spherical shell is half the kinetic energy at the 
instant of maximum velocity. Hence the total energy transmitted 



SPHERICAL WAVES IN ELASTIC SOLID. 211 

per period is equal to the wave-length multiplied into the surface- Molar, 
integral, over the whole spherical surface, of the maximum kinetic 
energy at any point per unit of volume : and therefore the energy 
transmitted per unit of time is equal to the product of the propa- 
gational velocity into this surface-integral. Thus we find that the 
rates per unit of time of the transmission of energy by the two 
sets of waves, of amplitudes represented by (67), are respectively 
as follows: 



(equivoluininal) ,y p/<v/ 2 o> 2 ( K* + L 

o 



(irrotational) " ph yw 9 [(3 - 2 A') 2 + 4L-] v 



(68). 



28. The sum of thess two formulas is the whole rate of 
transmission of energy per unit of time, and must be equal to 
the average rate of doing work by the vibrating rigid globe upon 
the surrounding elastic solid. Hence if w denote this rate, we 
must have 

w = 2 | /r</'or {2 ( K* + L*) + v [(3 - 2A') 3 + 4L 2 ]] . . .(69). 

29. To verify this proposition, let us first find the resultant 
force, P, with which the globe presses and drags the elastic 
solid, and then the integral work which P does per period, and 
thence the average work per unit time. Going back to 9, we 
see that P is the surface-integral of X over the spherical surface 
of radius q. Hence by the first of equations (25), which, in virtue 
of the equations 

v*; ?i = pw 3 ............... (70), 



we may write as follows : 

X = p{CjSv*-[-2\*(A+C 3 ) + 2(2\ 3 + l)B + (\*+l)C 1 ]u*l ...(71), 

we find 



, _ 2 , 

3 \qv + q* 'W 

= 9 I 2 (v-u)^\ + 2(^-uvf 1 2 + 3^+ <4 ...... (72), 

q ( f qu* '3*1*' q ) 

142 



212 LECTURE XIV. 

Molecular, where Q denotes ir<fp, being the mass which our rigid globe 
would have if its density were equal to that of the elastic solid. 
Hence by (65), for simple harmonic motion in period Zir/ta, 

x^-SL " --" + 3 !" . si,, * 



or, substituting the values of K and L from (65), and denoting by 
D the common denominator, 



- v" sin tat 



" J) 



I : -. (n- + 2v- + q-ar) - \("-- v-)" + , (4w - v") si 
(\_q (D q & q~o>~ 

\~2it /tiv* 3w 2 A 

+ 4 4+ - - + l 

| f/w V<jro> 7-0)- / 



This may be written for brevity 

P = k (<i sin wt + b cos wt) (72'"). 

Finally for w we have, denoting the period by r, 

I C T l [ T 

w = - I dtPf (t) = Ji-bco I dt cos- wt = IfJrbw (73) 

TJO rJo 

' Q W K Cw + ,J + l } + \ C + ,J + 1 }] ' - (7:i ' ) - 

LJ X J J J X J j 7 _| 

30. To verify the agreement of this direct formula for the 
work done, with ((59) which expresses the effect produced in 
waves travelling outwards at great distances from the centre, 
is a very long algebraical process, with K and L in (69) given 
by (65). But it becomes very simple by the aid of the following 
modified formulas for J^ (t) and J/C (t), which are also useful 
for other purposes. From (48), (49), and (50), by eliminating 
J?i, we get an equation for J/C similar to (49), viz., 

Id- + l (u + 2v) d + - (i 
L q <f 

where 



SPHERICAL WAVES IN ELASTIC SOLID. 213 

We may now write (50) in the form Molecular. 

hqu'G sin (tat + a), 



and similarly 



.(75), 



where the values of G, H, a, /9, are given by the following 
equations : 

G cos a = 1 ; G sin a = | 
q to (jo) 

^ - (76). 

/fcos = l- //"sin/9= 
qW qto i 

From the second of equations (53) we therefore have 
<r / \ _ ; -r ^ sm ( w t + a ) ~~ N cos (a>t + a) 

and 



where J/- and iV- denote the terms of the denominator of the 
thin! of equations (53). By the same method of investigation 
as that which gave us (69), we now find for the sum of the rates 
of transmission of energy 



Substituting the values of G, If, M, N, and introducing the 
notation Q, we obtain 



-,,(' + 2,')- ij +, 



ifiii (79 , 



This agrees with the value of w given by (73') ; and thus the 
verification is complete. 

31. In (78) the numerator of the last factor shows the 
parts due to the equivoluminal and irrotational waves respectively. 
Denoting by ./ the ratio of the energy of the equivoluminal wave 
to that of the irrotational, we have 



2w i ^- ^ , , 

.(80). 



v( -/.+ 2 U .+ 1) 

V&> 4 </ 2 <u- / 



214 



LECTURE XIV. 



32. Consider the following four cases : 

(a) qca very large in comparison with the larger 
of u and v. 



(6) qw = v. 



J 



. 8 

U V V 



...(81). 



(d) qw very small in comparison with the smaller 
of u and v. 



If v=x> , cases (a) and (b) cannot occur; and in cases (c), (d) 
we see by ((Si) that J= s> ; that is to say, the whole energy is 
carried away by the equivoluminal waves. If v is very small in 
comparison with u, we find that although J is infinite in cases (a) 
and (c), it is zero in cases (6) and (d). This to my mind utterly 
disproves my old hypothesis* of a very small velocity for irrotational 
wave-motion in the undulatory theory of light. 

33. Let us now work out some examples such as that 
suggested in an addition of March 6, 1899, to this Lecture 
(p. 188), but witli the simplification of assuming a rigid massless 
spherical lining for the cavity, which for brevity I shall call the 
sheath. But first let us work out in general the problem of 
finding what force in simple proportion to velocity must be 
applied to a mass in mounted on massless springs as described 
in p. 145 above, to keep the sheath vibrating in simple harmonic 
motion /( sin wt, and therefore to do the work of sending out the 
two sets of waves with which we have been concerned. Let 7 
be the required force per unit of velocity of in ; so that 7^ is 
the working force that must be applied to m, at any time when 
e is its displacement from its mean position. Now the springs, 
which must act on the sheath with the force P of (72') above, 
must react with an equal force on in because they are massless ; 

* " On the Reflexion and Infraction of Light," Phil. May. 1888, 2nd half year. 



SPHERICAL WAVES IN ELASTIC SOLID. 215 

so that the equation of motion of in is Molar. 



And, by the law of elastic action of the springs, we have 

P = c(e-hsma)t) ..................... (83), 

where c denotes what I call the "stiffness " of the spring-system. 

34. For e, (83) and (72'") give 

e = h \(l -f ")sina>*+-cosa>n ........... (84); 

and with this in (82) we find 
war ( 1 + J sin (at -f - cos cot 

= f a + yto - J sin o>t +\b yco ( 1 + - ) sin tot, 
which requires that 

(\ b ,6 i /, A 

1 + - ) 7/to) 2 = + 7>, and - mar = o 1 1 + - ] yw ; 
c/ c c \ c / 

by which, solved for two unknown quantities, <yto and into-, we 
find 



and 



If we suppose o> and c known, these equations, with (72"), (72'") 
for a and b, tell what m/'Q must be in order that the force 
applied to maintain the periodic motion of the sheath shall vary 
in simple proportion to the velocity ; and they give 7, the mag- 
nitude of this force per unit velocity. 

35. If we denote by E the maximum kinetic energy of in, 
we find immediately from (84), 



And by (73) we have, for the work per period done on the sheath 
byP, 

TW = Th*tob ........................... (88). 



216 



LECTURE XIV. 



Molecular. This ought to agree with the work done by ye per period, being 
j r dte 7 e, which, by (84), is faM |~(l + fj" + (7)"] ...(89). 



w 1 + - 

LV c 
The agreement between (89) and (88) is secured by (85). 

36. By (87), (89), and ro> = 2-rr] and by (86), (85), we find 
E m _ mar _ a (a + c) + b 2 
TW Ty 2Tryo) 2?r6c 

which, as we shall see, is a very important result, in respect to 
storage of energy in vibrators for originating trains of waves. 

37. Remark now that a, b, c are each of the dimensions 
of a " longitudinal stiffness," that is to say Force -f- Length, or 
Mass -T- (Time) 2 ; and for clearness write out the full expressions 
for a and b from (72") and (72'") as follows; 



a = Qo)- 



u- + '2v- _ 1 \- in + 2iA- 



- . + - = + 1 + - 




...(91). 



38. Let qw be very large in comparison with the larger of u 
or v (Case I. of 32). We have 

~ , 4>uv v~ 7 ~ v 
a == Qw- ; b = Qw ; 



therefore , = ; = (92). 

b ' ' TW ' 2?rc 

This case is interesting in connection with the dynamics of 
waves in an elastic solid, but not as yet apparently so in respect 
to light. 

39. Let qw be very small in comparison with the smaller of 
u or v (Case IV. of 32). We have 



therefore 



E " ( + c > 



SPHERICAL WAVES IN ELASTIC SOLID. 217 

This case is supremely important in respect to molecular sources Molar, 
of light. 

40. Let c be very small in comparison with a + 6 8 /a. We 
have 

E . a*b + b be* 



41. Letc=x. We have 

= a - <o = b- - = a (95) 

rw 27T& ' 7&> 

This is the simple case of a rigid globe of mass m embedded 
firmly in the elastic solid, and no other elasticity than that of the 
solid around it brought into play. It is interesting in respect to 
Stokes' and Rayleigh's theory of the blue sky. 

42. Let v = ao . We have 
u s 1 



This case is of supreme interest and importance in respect to 
the Dynamical Theory of Light. 

43. Take now the particular example suggested in the 
addition of March 6, 181)!) (p. 188 above), which is specially 
interesting as belonging to cases intermediate between those of 
38 and 39 ; a vast mass of granite with a spherical hollow of 
ten centimetres diameter acted on by an internal simple harmonic 

vibrator of 1006 periods per second (being 1000 */") Tnis 

makes o> 2 = 40 x 10", qW = 10, <w = 6324, qco = 31620. Now the 
velocities of the equivoluminal and the irrotational waves in 
granite* are about 2'2, and 4 kilometres per second; so we have 
u = 2-2 x 10 s , v = 4 x 10 s . Hence, and by (80) and (91), 



W = 0-957 ; - , = 48-4 ; ~ = 2342'56 ; 
/&) q^ta- (fa) 4 

- =12-649; -""-,= 160; ** =25600; 
qo) ' 9 J <o 2 q 4 to 4 

a=189-lxQo) 8 ; b = 25'59 x Qo>'; ^ = 
Gray and Milne, Phil. Mag., Nov. 1881. 



...(97). 



218 LECTURE XIV. 

Molecular. And by (85), (86), (90) ; with for brevity c = sQw", 

25-59. s" n w_ [189-1 (189-1 + s)4-654-8], 

7&> = (189-1 + s) 2 + 654-8 X ^' ' Q ~ (189-f+ s)' + 654'8 
E = 7^90 (189-1 -f s) + 25-59 = ri7c + 226 

TW 2-7TS * 

(98). 

44. As a first sub-case take ( 41) c = oo : we find by (95), 
(97), w= 189'lQ; and A7rw= T176. These numbers show that 
the kinetic energy of m at each instant of transit through its 
mean position, supplies only 1*176 of the energy carried away in 
the period by the outward travelling waves ; though its mass is 
as much as 189 times that of granite enough to fill the hollow. 
Hence we see that if the moving force ye were stopped the motion 
of m would subside very quickly and in the course of six or seven 
times r it would be nearly annulled. The not very simple law of 
the subsidence presents an extremely interesting problem which is 
easily enough worked out thoroughly according to the methods 
suitable for 25 above. Meantime we confine ourselves to cases 
in which E/rw is very large. 

45. Such a case we have, under 39, 41, if instead of 1006 
periods per second we have only T0065; which makes </ 2 or = 1000; 
tjta = 10 V10 = 31-620 ; and, by (93), (95) with s-'till our values of u 
and v for granite, 

TT 

a=l-892xlO"x Qo> 2 ; ^ = 7394; -^ = 1177 (99). 

Hence the kinetic energy of 111 in passing through the middle 
of its range is nearly 1200 times the work required to maintain 
its vibration at the rate of 1 0005 periods per second : and the 
value found for a, used in (95), shows that ///, supposed to be a 
rigid globe filling the hollow, must be 189 million times as dense 
as the surrounding granite, in order that this period of vibration 
can be maintained by a force in simple proportion to velocity. 
(See 40.) It is now easy to see that if the maintaining force 
is stopped, the rigid globe will go on vibrating in very nearly the 
same period, but subsidentially according to the law 

-f/r 9 . 

7/6'^ sin ' (100); 



SPHERICAL WAVES IN ELASTIC SOLID. 219 

and there will be the corresponding subsidence in the amplitudes Molecular. 
of the two sets of waves travelling outwards in all directions at 
great distances from the origin, when, according to the pro- 
pagational velocities u, v, the effects of the stoppage of the 
maintaining force reach any particular distance. 

It is quite an interesting mathematical problem, suggested at 
the end of 44, to fully determine the motion in all future time, 
when 7/1 is left with no applied force, after any given initial 
conditions, with any value, large or small, of mjQ. 

4G. Returning now to the maintenance of vibrations at 
the rate of 1000^ periods per second; and u, v for granite; and 
9 = 5 cm., all as in 4.3 ; see (98) and remark that, to make 
JS/TW very large, * must be a very small fractional numeric ; and 
this makes m/Q =8. To take a vibrator not differing greatly in 
result from the violin string suggested in the addition of March 6 
(p. 18S), let in be a little ball of granite of cm. diameter. This 
makes m = Q/KOOO, and therefore (40) *= 1/8000. Hence by 
(98), E/-nv= 1808001, from which, with what we know of wave- 
motion, we infer that if in be projected with any given velocity, V, 
from its jxxsition of equilibrium, it will for ever after vibrate, 
with amplitude diminishing according to the formula 



(101). 



Thus during 3,610,002 periods the range of m will be reduced in 
the ratio of 6 to 1 (say approximately 2 to 1), by giving away its 
energy to be transmitted outwards by the two species of waves, 
of which, according to J of (97), the equivoluminal takes twelve 
times as much as the irrotational.] 



LECTURE XV. 

TUESDAY, October 14, 5 P.M. 

Molecular. RETURNING to our model, wo shall have in a short time a state 
of things not very different from simple harmonic motion, if we 
get up the motion very gradually. We have now an exciting 
vibration of shorter period than the shortest of the natural 
periods. We must keep the vibrator going through a uniform 
range. We are not to augment it, and it will be a good thing 
to place something here to mark its range. [This done.] Keep it 
going long enough and we shall see a state of vibration in which 
each bar will be going in the opposite direction to its neighbour. If 
we keep it going long enough we certainly will have the simple 
harmonic motion ; and if this period is smaller than the smallest 
of the three natural periods, we shall, as we know, have the alternate 
bars going in opposite directions. Now you see a longer-period 
vibration of the largest mass superimposed on the simple har- 
monic motion we are waiting for. I will try to help towards 
that condition of affairs by resisting the vibration of the top 
particle. In fact, that particle will have exceedingly little motion 
in the proper state of things (that is to say, when the motion is 
simple harmonic throughout), and it will be moving, so far as it 
has motion at all, in an opposite direction to the particle im- 
mediately below it. It is nearly quit of that superimposed 
motion now. We cannot give a great deal of time to this, but 
I think we may find it a little interesting as illustrating 
dynamical principles. Prof. Mendenhall is here acting the part 
of an escapement in keeping the vibrator to its constant range. 
W T e cannot get quit of the slow vibration of the particle. A 
touch upon it in the right place may do it. A very slight touch 



EXCITATION OF SYNCHRONOUS VIBRATOR BY LIGHT. 221 

is more than enough. I have set it the wrong way. Now we Molecular, 
have got quit of that vibration and you see no sensible motion 
of w, at all. These two, (,, w,), .are going in opposite directions, 
and the lower one in opposite direction to the exciter. There- 
fore this is a shorter vibration than the shortest natural period. 
Now I set it to agree with the shortest of the periods, the first 
critical position. If we get time in the second lecture to-day, 
I am going to work upon this a little to try to get a definite 
example illustrating a particle of sodium. Before we enter upon 
any hard mathematics, let us look at this a little, and help our- 
selves to think of the thing. What I am doing now is very 
gradually getting up the oscillation. I am doing to that system 
exactly what is done to the sodium molecule, for example, when 
sodium light is transmitted through sodium vapour. We may feel 
quite certain, however, that the energy of vibration of the 
sodium molecule goes on increasing during the passage through 
the medium of at least two-hundred thousand waves, instead 
of two dozen at the most perhaps that I am taking to got up 
this oscillation. But just note the enormous vibration we have 
here, and contrast it with the state of things that we had just 
before. The upper particle is in motion now and is performing 
a vibration in the same period and phase as the lower particle, 
only through comparatively a very small range. The second 
particle, I am afraid, will overstrain the wire. (By hanging up a 
watch, bifilarly, so that the period of bifilar suspension approxi- 
mately agrees with the balance wheel, you get likewise a state 
of wild vibration. But if you perform such experiments with a 
watch, you are apt to damage it.) This, which you see now, is 
a most magnificent contrast to the previous state of things when 
the period of the exciter was very far from agreeing with any of 
the fundamental periods. 

We will now return to our molar subject, the clastic solid. 
You will see a note in the paper of yesterday to which I have 
referred, stating that the thlipsinomic method is more convenient 
than the tasinomic for dealing with incompressibility, and in 
point of fact it is so. 

I explained to you yesterday Rankine's nomenclature of 
thlipsinomic and tasinomic coefficients, according to which, when 
the six stress-components are expressed in terms of strain- 
components, the coefficients are called tasinomic; and when the 



222 LECTURE XV. 

Molar. strain-components arc expressed in terms of the stress-components, 
the coefficients are called thlipsinomic. [Thus, going back to 
Lecture XI. (p. 132 above), we see, in the six equations (1), 36 
tasinomic coefficients expressing the six stress-components as 
linear functions of the six strain-components : and we sec, in 
virtue of 15 equalities among the 3G coefficients, just 21 indepen- 
dent values, being Green's celebrated 21 coefficients. Use now 
those six equations to determine the six quantities e,f, g, a, b, c in 
terms of P, Q, R, S, T, U. Thus we find 

e = (PP) P 4 (PQ) Q + (PR) R + (P#) tf + (PT) T + (P U) U 
f = (QP) P + (QQ) Q + (QR) R + (QS) S + (QT) T + (Q U) U 
fj = (RP) P + (RQ) Q + (RR] R + (RS) 8 + (RT) T+ (RU) U 
a = (UP) P + (SQ) Q + (SR) R + (SS) S + (ST) T + (SU) U 
b = (TP) P + (TQ) Q + (TR) R + (TO) S + (TT) T + (TU) U 
c = (UP)P + (UQ) Q + (UR) R + (Uti)S + (UT) T + (UU) U 

0), 

where (PP), (PQ), &c., denote algebraic functions of 11, 12, 22, 
&c., found by the process of elimination. This process, in virtue 
of the 15 equalities 12 = 21, &c., gives (PQ} = (QP\ &c.; 15 
equalities in ail. The 21 independent coefficients (PP), (PQ), &c., 
thus found, are what Ratikine called the thlipsinomic coefficients. 
Taking now from Lecture II., p. 24, 

E = $(Pe+Qf+Rg + Sa+ Tb+ Uc) (2), 

and eliminating P, Q, R from this formula by the equations (1) 
of Lecture XL, (p. 132), we find the tasinomic quadratic function 
for the energy which we had in Lecture II. (p. 23). And eliminat- 
ing e, f, g, a, b, c from it by our present six equations, we find the 
corresponding thlipsinomic formula for the energy with 21 inde- 
pendent coefficients (PP), (PQ), &c.*] In a certain sense, these 
coefficients, both tasinomic and thlipsinomic, may be all called 
moduluses of elasticity, inasmuch as each of them is a definite 
numerical measurement of a definite elastic quality. I have, 
however, specially defined a modulus as a stress divided by a 
strain, following the analogy of Young's modulus. If we adhere 
to this definition, then the tasinomic coefficients are moduluses, 
and the thlipsinomic coefficients are reciprocals of moduluses. 

* Compare Thomson and Tait, 673, (12) (20). 



THLIPSINOMIC TREATMENT OF COMPRESSIBILITY. 223 

In the Lecture notes in your hands for today you see the Molar. 
thlipsinomic discussion of the question of compressibility or in- 
compressibility ; which is much simpler than our tasinomic 
discussion of the same subject in which we failed yesterday. You 
see that if the dilatation, e +f+g, be denoted by 8, you have 

8 = [(PP) + (QP) + (RP)] P + [(PQ) + (QQ) + (RQ)] Q 
+ [(PR) + (QR) + (RR)] R + [(PS} + (QS) + (RS)] S 
+ [(PT) + (QT) + (RT)]T + [(PU) + (QU) + (RU)]U ...... (3). 

Thus if P is the sole stress, a dilatation [(PP) + (QP) + (RP)] P 
is produced ; and if S is the sole stress, a dilatation 



is produced. We see therefore that S', a kind of stress which in 
an isotropic solid would produce merely distortion, may produce 
condensation or rarefaction in an a?olotropic solid. The coefficients 
of P, Q, R, S, T, U in our equation for 8 may be called compressi- 
bilities. Their reciprocals are (according to my definition of a 
modulus) motluluses for compressibility. In an isotropic solid, 
each of the last three coefficients vanishes ; and the reciprocal of 
each of the others is three times what I have denoted by k 
(Lecture II., p. 25), and called the compressibility-modulus or the 
bulk-modulus, being P/B, where P denotes equal pull, or negative 
pressure, in all directions. 

An iKolotropic solid is incompressible if, and is compressible 
unless, each of the six coefficients in our formula for 8 vanishes. 
That is to say, it is necessary and sufficient for incompressibility 
that 

(PP) + (QP) + (RP) 



+(QS)+(RS)=Q 
(PT) + (QT)+(RT) = Q 
(QU)+(RU)=0 



Thus we see that six equations among the 21 coefficients suffice 
to secure that there can be no condensational-rarefactional wave, 
or, what is the same thing, that, in every plane wave, the vibration 
must be rigorously in the plane of the wave-front; and therefore 
that Green was not right when, in proposing to confine himself 



224 LECTURE XV. 

Molar. " to the consideration of those media only in which the directions 
of the transverse vibrations shall always be accurately in the 
front of the wave," he said*, "This fundamental principle of 
Fresnel's theory gives fourteen relations between the twenty-one 
constants originally entering into our function." What Green 
really found and proved was fourteen relations ensuring, and 
required to ensure, that there can be plane waves with direction 
of vibration accurately in the plane of the wave, if there can also 
be condensational-rarefactional waves in the medium. What we 
have now found is that not fourteen, but only six, equations suffice 
to secure incompressibility and therefore to compel the direction 
of the vibration in any actual plane wave to be accurately in the 
plane of the wave. 

[March 7, 1899. I have only today found an interesting and 
instructive mode of dealing with those six equations of incom- 
pressibility, which I gave in this Baltimore Lecture of October 14, 
1884. By the first three of them, eliminating (PP), (QQ), (RR), 
and by the other three, (PS), (QT), (RU) from the equation of 
energy, we find 

2E = -[(QR)(Q-Ry + ( RP) (R - P) 2 + ( PQ) (P - QT~] 

+ 2{[(QU)U-(RT)T](Q-R)+[(RS)S-(PU)U](R~P) 

+ [(PT)T-(QS)8](P-Q)} 
+ (SS) 8* + (TT) T- + ( UU} U* 
+ 2[(TU)TU+(US)US + (8T)ST] ..................... (5). 

In this, P, Q, R appear only in their differences, which is an 
interesting expression of the dynamical truth that if P = Q = R, 
they give no contribution to the potential energy. 

The three differences Q R, R P, P Q, are equivalent to 
only two independent variables ; thus if we put P Q = V and 
P-R= W, we have Q - R = W - V, and the expression for E 
becomes a homogeneous quadratic function of the five indepen- 
dents V, W, S, T, U, with fifteen independent coefficients, which 
we may write as follows : 



+ (VW)VW +(VS)VS 

+ (WS)WS +(WT)WT+(WU)WU 

+ (ST)ST + (SU)SU +(TU)TU ........................ (G). 

* Green, Collected Papers, p. 293. 



THLIPSINOMIC TREATMENT OF COMPRESSIBILITY. 225 

The differential coefficients of this with reference to S, T, U Molar, 
are of course as before, a, 6, c. Denote now by h and i its 
differential coefficients with reference to V and W. Thus we find 

h = (VV) V+ (VW)W+ (VS)S+ (VT)T+ (VU)U 
' + (WS)S + (WT)T + (WU)U 
(SW)W+ (88) S+ (ST)T+ (SU)U -..(7); 
6= (TV)V+ (TW)W+ (TS)S+ (TT)T + (TU)U 
c= (UV)V+(UW)W + (US)S + (UT)T+(UU)U 
and we have 

The dynamical interpretation shows that h must represent f, 
and t must represent g, when V and W represent P Q, 
and P-R. 

Solving the five linear equations (7) for V, W, S, T, U, we 
find the tasinomic expressions for the five stress-components in 
terms of the five strain-components, which we may write as 
follows : 

V = (hh) h + (hi) i + (ha) a + (hb) b + (he) c\ 

W = (ih) h + (U) t + (ia) a + (ib) b + (ic) c\ 

S = (ah)h + (ai)i + (aa)a + (ab)b + (ac)c\ (9). 

T = (bh) h + (bi) i + (ba) a + (bb) b + (be) c I 

U = (ch) h + (ci) i + (ca) a + (cb) b + (cc) c) 
The algebraic process shows us that 

(hi) = (ih)-, (ha) = (ah); &c (10); 

so that we have now found the 15 independent tasinomic coeffi- 
cients from the 15 thlipsinomic. Lastly, eliminating by (9) 
V, W, S, T, U from (8), we find the tasinomic quadratic expressing 
the energy.] 

As I said in the first lecture, one fundamental difficulty is 
quite refractory indeed. In the wave-theory of light the velocity 
of the wave ought to depend on the plane of distortion. If you 
compare the details of motion in the wave-surface* worked out 
for an incompressible a3olotropic elastic solid, with equalities 
enough among the coefficients to annul all skewnesses, you will 
see that it agrees exactly with Fresnel's wave-surface but that 
instead of the direction of the line of vibration of the particles as 

* Lecture XIL, p. 137; Lecture XIII., p. 175. 
T. L. 15 



226 



LECTURE XV. 



Molar. in Fresnel's construction we have the normal to the plane of distor- 
tion as the direction on which the propagational velocity depends. 
I see no way of getting over the difficulty that the return 
forces in an elastic solid the forces on which the vibration 
depends are dependent on the strain experienced by the solid 
and on that alone. I have never felt satisfied with the ingenious 
method by which Green got over it. Stokes quotes in his report 
on Double Refraction, page 265 (British Association 1862): "In 
" his paper on Reflection, Green had adopted the supposition of 
"Fresnel that the vibrations are perpendicular to the plane of 
" polarization. He was naturally led to examine whether the laws 
" of double refraction could be explained on this hypothesis. 
" When the medium in its undisturbed state is exposed to pressure 
" differing in different directions, six additional constants are intro- 
" duced into the function </>, or three in the case of the existence of 
" planes of symmetry to which the medium is referred. For waves 
" perpendicular to the principal axes, the directions of vibration 
" and squared velocities of propagation are as follows : 



Wave normal 


X 


>.) 




Direction of vibration 


X 


G + A 


N + B 


M+C 


y 

z 


N + A 


H+B 


L + C 


M + A 


L + B 


I + C 



"Green assumes, in accordance with Fresnel's theory, and with 
"observation if the vibrations in polarized light are supposed 
" perpendicular to the plane of polarization, that for waves 
" perpendicular to any two of the principal axes, and propagated 
" by vibrations in the direction of the third axis, the velocity of 
" propagation is the same." 

Let us see what this statement means before considering 
whether it may be verified, as Green supposes, by the introduction 
of " extraneous pressure." Consider waves having their fronts 
parallel to the sides N and W (North and West) of this box, which 
are perpendicular to two of the three principal axes of the crystal, 
and such having its vibrations in the direction of the third axis (up 
and down). Take first the wave that is propagated south as I hold 






WAVE VELOCITY DEPENDS ON LINE OF VIBRATION. 



227 



the box. There is the plane of the wave (N). The vibration up Molar. 
and down with N held fixed will give a shear like that marked 1, 




in which a square becomes a rhombic figure. That represents the 
strain in the solid corresponding to this first state of motion. Simi- 
larly the wave propagated in the eastward direction will give rise 
to a shear of this kind marked 2, the vibration still being upward. 
The assumption is that one of these sets of waves is propagated 
at the same speed as the other. That is to say, the waves which 
have their shear in this west plane have the same velocity as 
the waves which have their shear in this north plane. The 
essence of our elastic solid is three different rigidities, one for 
shearing in this plane W, one for shearing in this plane N, and 
one for shearing in the other principal plane (the horizontal plane 
of our box). The incongruous assumption is that the velocities of 
propagation do not depend on the planes of the shearing strain, 
and do depend, simply and solely, on the direction of the vibration. 
The introduction by Green (in order to accomplish this) of 
what he calls " extraneous force," which gives him three other 
coefficients has always seemed to me of doubtful validity. In 
the little table above, taken from Stokes, L, M, N are the three 
principal rigidities, the 44, 55, 66 of our own notation. A, B, C 
are the effects of extraneous pressure. The table gives the squared 
velocities of propagation and waves of different wave-normals and 
directions of vibration along the axes. The principal diagonal refers 
only to condensational waves, or waves in which the direction 
of vibration coincides with the wave-normal. Green's assumption 

152 



228 LECTURE XV. 

Molar. makes, for vibrations in the ^'-direction, N + B = M+C', which, 
with the two corresponding equations for vibrations in the 
directions y, z, gives 

A-L = B-M=C-N. 

[March 7, 1899. ADDITION. ON CAUCHY'S AND GREEN'S 
DOCTRINE OF EXTRANEOUS FORCE TO EXPLAIN DYNAMICALLY 
FRESNEL'S KINEMATICS or DOUBLE REFRACTION*. 

1. Green's dynamics of polarization by reflexion, and Stokes' 
dynamics of the diffraction of polarized light, and Stokes' and 
Rayleigh's dynamics of the blue sky, all agree in, as seems to me, 
irrefragably, demonstrating Fresnel's original conclusion, that in 
plane polarized light the line of vibration is perpendicular to the 
plane of polarization; the "plane of polarization" being defined 
as the plane through the ray and perpendicular to the reflecting 
surface, when light is polarized by reflexion. 

2. Now when polarized light is transmitted through a crystal, 
and when rays in any one of the principal planes are examined, it 
is found that 

(1) A ray with its plane of polarization in the principal 
plane travels with the same speed, whatever be its direction 
(whence it is called the " ordinary ray " for that principal plane) ; 
and (2) A ray whose plane of polarization is perpendicular to the 
principal plane, and which is called the " extraordinary ray " of 
that plane, is transmitted with velocity differing for different 
directions, and having its maximum and minimum values in two 
mutually perpendicular directions of the ray. 

3. Hence and by 1, the velocities of all rays having their 
vibrations perpendicular to one principal plane are the same ; and 
the velocities of rays in a principal plane which have their direc- 
tions of vibration in the same principal plane, differ according to the 
direction of the ray, and have maximum and minimum values for 
directions of the ray at right angles to one another. But in the 
laminar shearing or distortional motion of which the wave-motion 
of the light consists, the "plane of the shearf" (or "plane of 
the distortion," as it is sometimes called) is the plane through the 

* Reprinted, with additions, from the Proc. R. S. E., Vol. xv. 1887, p. 21, and 
Phil. Mag., Vol. xxv. 1888, p. 116. 

t Thomson and Tait's NaturaJ Philosophy, 171 (or Elements, 150). 



CAUCHY AND GREEN'S EXTRANEOUS FORCE. 229 

direction of the ray and the direction of vibration ; and therefore Molar, 
it would be the ordinary ray that would have its line of vibration 
in the principal plane, if the ether's difference of quality in 
different directions were merely the aeolotropy of an unstrained 
elastic solid*. Hence ether in a crystal must have something 
essentially different from mere intrinsic seolotropy ; something 
that can give different velocities of propagation to two rays, of 
one of which the line of vibration and Hue of propagation coincide 
respectively with the line of propagation and line of vibration 
of the other. 

4. The difficulty of imagining what this something could 
possibly be, and the utter failure of dynamics to account for double 
refraction without it, have been generally felt to be the greatest 
imperfection of optical theory. 

It is true that ever since 1839 a suggested explanation has 
been before the world ; given independently by Cauchy and 
Green, in what Stokes has called their " Second Theories of 
Double Refraction," presented on the same day, the 20th of 
May of that year, to the French Academy of Sciences and the 
Cambridge Philosophical Society. Stokes, in his Report on Double 
Re fraction f, has given a perfectly clear account of this explana- 
tion. It has been but little noticed otherwise, and somehow it 
has not been found generally acceptable ; perhaps, because of a 
certain appearance of artificiality and arbitrariness of assumption 
which might be supposed to discredit it. But whatever may have 
been the reason or reasons which have caused it to be neglected 
as it has been, and though it is undoubtedly faulty, both as given 
by Cauchy and by Green, it contains what seems to me, in all 
probability, the true principle of the explanation, and which is, 
that the ether in a doubly refracting crystal is an elastic solid, 
unequally pressed or unequally pulled in different directions, by 
the unmoved ponderable matter. 

5. Cauchy's work on the wave-theory of light is complicated 
throughout, and to some degree vitiated, by admission of the 

* The elementary dynamics of elastic solids shows that on this supposition 
there might be maximum and minimum velocities of propagation for rays in 
directions at 45 to one another, but that the velocities must essentially be equal for 
every two directions at 90 to one another, in the principal plane, when the line of 
vibration is in this plane. 

t British Association Report, 1862. 



230 LECTURE XV. 

Navier-Poisson false doctrine* that compressibility is calculable 
theoretically from rigidity ; a doctrine which Green sets aside, 
rightly and conveniently, by simply assuming incompressibility. 
In other respects Cauchy's and Green's " Second Theories of 
Double Refraction," as Stokes calls them, are almost identical. 
Each supposes ether in the crystal to be an intrinsically seolotropic 
elastic solid, having its seolotropy modified in virtue of internal 
pressure or pull, equal or unequal in different directions, produced 
by and balanced by extraneous force. Each is faulty in leaving 
intrinsic rigidity-moduluses (coefficients) unaffected by the equi- 
librium-pressure, and in introducing three fresh terms, with 
coefficients (A, B, C in Green's notation) to represent the whole 
effect of the equilibrium-pressure. This gives for the case of an 
intrinsically isotropic solid, augmentation of virtual rigidity, and 
therefore of wave- velocity, by equal pullf in all directions, and 
diminution by equal positive pressure in all directions ; which is 
obviously wrong. Thus definitively, pull in all directions outwards 
perpendicular to the bounding surface equal per unit of area to 
three times the intrinsic rigidity-modulus, would give quadrupled 
virtual rigidity, and therefore doubled wave-velocity ! Positive 
normal pressure inwards equal to the intrinsic rigidity-modulus 
would annul the rigidity and the wave-velocity that is to say, 
would make a fluid of the solid. And, on the other hand, nega- 
tive pressure, or outward pull, on an incompressible liquid, would 
give it virtual rigidity, and render it capable of transmitting 
laminar waves ! It is obvious that abstract dynamics can show 
for pressure or pull equal in all directions, no effect on any physical 
property of an incompressible solid or fluid. 

6. Again, pull or pressure unequal in different directions, on 
an isotropic incompressible solid, would, according to Green's 
formula (A) in p. 303 of his collected Mathematical Papers, cause 
the velocity of a laminar wave to depend simply on the wave- 
front, and to have maximum, minimax, and minimum velocities 

* See Stokes, "On the Friction of Fluids in Motion and on the Equilibrium and 
Motion of Elastic Solids," Camb. Phil. Trans., 1845, 19, 20; reprinted in Stokes' 
Mathematical and Physical Papers, Vol. i. p. 123 ; or Thomson and Tait's Natural 
Philosophy, 684, 685; or Elements, 655, 656. 

t So little has been done towards interpreting the formulas of either writer that 
it has not been hitherto noticed that positive values of Cauchy's O, H, I, or of 
Green's A, B, C, signify pulls, and negative values signify pressures. 



STRESS-THEORY OF DOUBLE REFRACTION. 231 

for wave-fronts perpendicular respectively to the directions of Molecular. 
maximum pull, minimax pull, and minimum pull; and would 
make the wave-surface a simple ellipsoid ! This, which would be 
precisely the case of foam stretched unequally in different direc- 
tions, seemed to me a very interesting and important result, until 
(as shown in 19 below) I found it to be not true. 

7. To understand fully the stress-theory of double refraction, 
we may help ourselves effectively by working out directly and 
thoroughly (as is obviously to be done by abstract dynamics) 
the problem of 6, as follows : Suppose the solid, isotropic 
when unstrained, to become strained by pressure so applied to 
its boundary as to produce, throughout the interior, homogeneous 
strain according to the following specification : 

The coordinates of any point M of the mass which were , 17, 
when there was no strain, become in the strained solid 

V, W& rV7 ..................... (1); 

\/o, V/9, v/7, r the " Principal Elongations*," being the same 
whatever point M of the solid we choose. Because of incompres- 
sibility we have 

a#y=l .............................. (2). 

For brevity, we shall designate as (a, ft, 7) the strained condition 
thus defined. 

8. As a purely kinematic preliminary, let it be required to 
find the principal strain -ratios when the solid, already strained 
according to (1), (2), is further strained by a uniform shear, tr, 
specified as follows ; in terms of jc, y, z, the coordinates of still the 
same particle, M, of the solid and other notation, as explained 
below : 



(3), 



where p = OP = X V + M V + *? \/7 ............... (4), 

with P+ro 8 +n=l, V + ^+i/^l ............... (5), 

and l\ + mn + nv = Q ........................ (6); 



See chap. iv. of "Mathematical Theory of Elasticity" (W. Thomson), Tram. 
Roy. Soc. Loud. 1856, reprinted in Vol. in. of Mathematical and Physical Paptrt, 
now on the point of being published, or Thomson and Tail's Natural Philosophy, 
| 160, 164, or EUmentt, 141, 158. 



232 LECTURE XV. 

X, fju, v denoting the direction-cosines of OP, the normal to the 
shearing planes, and I, m, n the direction-cosines of shearing dis- 
placement. The principal axes of the resultant strains are the 
directions of OM in which it is maximum or minimum, subject to 
the condition 

2 + ??2 + 2= l ........................ (7); 

and its maximum, minimax, and minimum values arc the three 
required strain-ratios. Now we have 



mrj V/3 + nf \/y)p + v^p 1 . . .(8), 
and to make this maximum or minimum subject to (7), we have 



- -... 

where in virtue of (7), and because OM' 2 is a homogeneous quad- 
ratic function of , rj, , 

p = OM*. 

The determinantal cubic, being 



= ...... (10), 

where 6/1 = o (1 + 2<rl\ + o- 2 X 2 ) ; ^= ft (1 -f 2<ri/u, + crV) 5 

< ^=7(l + 2o-/iy + o-V) .................. (11) 

and a = \/(/3y) [a (mv + nfi) + a~/J.v] ; 6 = \/(7) [<r (wX, + Zz/)+o- 2 z/\]; 
c=V(a/3)[o-(^ + m\) + (T 2 X/A] ............... (12), 

gives three real positive values for p, the square roots of which 
are the required principal strain-ratios. 

9. Entering now on the dynamics of our subject, remark 
that the isotropy (1) implies that the work required of the 
extraneous pressure, to change the solid from its unstrained 
condition (1, 1, 1) to the strain (a, /8, 7), is independent of the 
direction of the normal axes of the strain, and depends solely on 
the magnitudes of a, @, 7. Hence if E denotes its magnitude per 
unit of volume ; or the potential energy of unit volume in the 
condition (a, ft, 7) reckoned from zero in the condition (1, 1, 1) ; 
we have 

# = >K,/3,7) ........................ (13), 



STRESS-THEORY OF DOUBLE REFRACTION. 233 

where >/r denotes a function of which the magnitude is unaltered Molar. 
when the values of a, ft, 7 are interchanged. Consider a portion 
of the solid, which, in the unstrained condition, is a cube of unit 
side, and which in the strained condition (a, ft, 7), is a rectangular 
parallelepiped \/ ^ft Vy I Q virtue of isotropy and symmetry, 
we see that the pull or pressure on each of the six faces of this 
figure, required to keep the substance in the condition (a, ft, 7), is 
normal to the face. Let the amounts of these forces per unit area, 
on the three pairs of faces respectively, be A, B, C, each reckoned 
as positive or negative according as the force is positive pull, 
or positive pressure. We shall take 

A+B + C = ........................ (14), 

because normal pull or pressure uniform in all directions produces 
no effect, the solid being incompressible. The work done on any 
infinitesimal change from the configuration (a, ft, 7) is 



or (because afty = 1) , 



-A flfj ft-J 

10. Let Ba, Bft, By be any variations of a, ft, y consistent 
with (2), so that we have 

(a+ Ba) (ft + 80) (7 + By) = Ij , 

and afty=l 

Now suppose Ba, Bft, By to be so small that we may neglect their 
cubes and corresponding products, and all higher products. We 
have 

?? + 8 f + h + tSftSy + ftByBa + yBaSft = (17) ; 

a ft y 

/SaV /Bft By\* 
whence ( ) = ( -& H ) i 

whence, and by the symmetrical expressions, 

1 /" bP &V 1 



878 ^ (18). 



234 LECTURE XV. 

11. Now, if E + 8E denote the energy per unit bulk of the 
solid in the condition 

(a + 8a, (3 + 8/3, 7 + 87), 
we have, by Taylor's theorem, 



where H l} H 2 , &c. denote homogeneous functions of 8ot, 8/3, 87 of 
the 1st degree, 2nd degree, &c. Hence, omitting cubes, &c., and 
eliminating the products from H 2 , and taking H l from (15), we 
find 



where G, H, I denote three coefficients depending on the nature of 
the function \jr (13), which expresses the energy. Thus in (19), 
with (14) taken into account, we have just five coefficients inde- 
pendently disposable, A, B, G, H, I, which is the right number 
because, in virtue of a/3<y = 1, E is a function of just two indepen- 
dent variables. 

12. For the case of a = 1, /3 = I, 7 = 1, we have 

A=B=C=0 and G = H=I = G l , suppose ; 
which give 8 A' = G, (8a 2 + 8/3- + 87'-). 

From this we see that 26^ is simply the rigidity-modulus of the 
unstrained solid ; because if we make 87 = 0, we have 8a = 8/3, 
and the strain becomes an infinitesimal distortion in the plane 
(#?/), which may be regarded in two ways as a simple shear of 
which the magnitude is Sot* (this being twice the elongation in 
one of the normal axes). 

13. Going back to (10), (11), and (12), let a- be so small 
that o- 3 and higher powers can be neglected. To this degree 
of approximation we neglect abc in (10), and see that its three 
roots are respectively 

b 2 c- 



< 



* Thomson and Tait's Xatural Philosophy, 175, or Elements, 154. 




STRESS-THEORY OF DOUBLE REFRACTION. 



235 



provided none of the differences constituting the denominators is Molar, 
infinitely small. The case of any of these differences infinitely 
small, or zero, does not, as we shall see in the conclusion, require 
special treatment, though special treatment would be needed to 
interpret for any such case each step of the process. 

14. Substituting now for &l, &, <&, a, b, c in (20), their 
values by (11) and (12), neglecting a 3 and higher powers, and 
denoting by Sot, 8, 87 the excesses of the three roots above 
o, /3, 7 respectively, we find 



| X s 
[_ 



8/8 



7-0 

x 

P 



(nX + lv)* - -- 



-- - Q (mv + n/i) 
7 p 



87 = 7 \2<rnv + a 3 \ v* - ~(mv + n/i) s - -* (nX + liif\ \ 
P-y 0-7 



(21), 



and using these in (19), we find 
BE = <r (Al\ + BmfjL + Cnv) 

+ fa* \A\* + Bp* + Cv* 

+ L (mv + n/i) + M (n\ + lv? + 

+ 2a* (Gl*\* + 
where 

L _ 



+ wX)] 



-...(22), 






15. Now from (5) and (6) we find 

(mv + w/*) 8 = I - 1* - X 2 + 2 (PX - ;/A* - wV) (24), 

which, with the symmetrical expressions, reduces (22) to 

8 E = o- (A l\ + Bmp -f Cnv) + <r* {L + M + N 
i /n \T\ . 

...(25). 



16. To interpret this result statically, imagine the solid 
to be given in the state of homogeneous strain (a, 0, 7) through- 
out, and let a finite plane plate of it, of thickness h, and of very 
large area Q, be displaced by a shearing motion according to the 
specification (3), (4), (5), (6) of 8 ; the bounding-planes of the 



236 LECTURE XV. 

plate being unmoved, and all the solid exterior to the plate being 
therefore undisturbed except by the slight distortion round the 
edge of the plate produced by the displacement of its substance. 
The analytical expression of this is 



where /denotes any function of OP such that 
rh 

(27). 



If we denote by W the work required to produce the supposed 
displacement, we have 

h <W .................. (28), 



SE being given by (25), with everything constant except a-, a 
function of OP ; and J W denoting the work done on the solid 
outside the boundary of the plate. In this expression the first 
line of (25) disappears in virtue of (27); and we have 



+ (A- L) \* + (B- M) ^ + (C-N ) /- 

- LI* - Mm* - Nn* + 2 [(2G + L - M - N) IW }- -(29). 



(21 + N-L- M) wV]} j h 



When every diameter of the plate is infinitely great in comparison 
with its thickness, W/Q is infinitely small ; and the second 
member of (29) expresses the work per unit of area of the plate, 
required to produce the supposed shearing motion. 

17. Solve now the problem of finding, subject to (5) and 
(6) of 8, the values of I, m, n which make the factor { } of the 
second member of (29) a maximum or minimum. This is only 
the problem of finding the two principal diameters of the ellipse 
in which the ellipsoid 



....... (30) 

s cut by the plane 

(31). 



STRESS-THEORY OF DOUBLE REFRACTION. 237 

If the displacement is in either of the two directions (I, m, n) thus Molar. 
determined, the force required to maintain it is in the direction of 
displacement ; and the magnitude of this force per unit bulk of 
the material of the plate at any point within it is easily proved 
to be 



where {M} denotes the maximum or the minimum value of the 
bracketed factor of (29). 

18. Passing now from equilibrium to motion, we see at 
once that (the density being taken as unity) 

V*={M] ........................... (33), 

where V denotes the velocity of either of two simple waves whose 

wave- front is perpendicular to (X, /*, v). Consider the case of 

wave-front perpendicular to one of the three principal planes ; 

(yz) for instance: we have X=0; and, to make { } of (29) a 

maximum or minimum, we see by symmetry that we must either 

have 

(vibration perpendicular to principal plane) \ 

1=1, ?/i = 0, n = 0\ ...... (34). 

(vibration in principal plane). . .1 = 0, m = v, n = /J 
Hence, for the two cases, we have respectively : 

Vibration perpendicular to yz ) ,_. 

*] ' 



N + (B-M)n* + (C-N)S\ 
Vibration tn yz. . . F' = L + B^ + C v * + 4 (H + 1 - L) /*V . . .(36). 

19. According to Fresnel's theory (35) must be constant, and 
the last term of (36) must vanish. These and the corresponding 
conclusions relatively to the other two principal planes are satisfied 
if, and require that, 

A-L = B-M=C-N (37), 

and H + I=L; I+G = M-, G + H=N (38). 

Transposing M and N in the last of equations (37), substituting 
for them their values by (23), and dividing each member by 7, 
we find 

-^A = ^-A (39)- 

fry-*J3 yd-fr" 



238 LECTURE XV. 

whence (sum of numerators divided by sum of denominators), 

B-C = C-A = A-B 
7-a/3 aj3-(3y /3y-ya'" 

The first of these equations is equivalent to the first of (37) ; and 
thus we see that the two equations (37) are equivalent to one 
only ; and (39) is a convenient form of this one. By it, as put 
symmetrically in (40), and by bringing (14) into account, we find, 
with q taken to denote a coefficient which may be any function of 
(, ft 7) : 

A=q(S-/3y)- B = q(S-ya); C = q (S - a{3) 1 ( . 
where S = % (j3y + y + a/3) j '" 

and using this result in (23), we find 



or L = q (28 - /9 7 ) ; M = q (28 -ya); N= 

By (2) we may put (41) and (42) into forms more convenient for 
some purposes as follows : 



(45) ' 



Next, we find G, H, I ; by (38), (44), and (45) wo have 



whence, by (38) and (44), 



20. Using (43) and (47) in (19), we have 

] 

(48). 



STRESS-THEORY OF DOUBLE REFRACTION. 239 

Now we have, by (2), log (a/By) = 0. Hence, taking the variation Molecular. 
of this as far as terms of the second order, 



which reduces (48) to 



Remembering that cubes and higher powers are to be neglected 
we see that (50) is equivalent to 



Hence if we take q constant, we have 



Z + l + i-3] (52). 



It is clear that q must be stationary (that is to say, Sq = 0) for 
any particular values of a, , 7 for which (52) holds ; and if (52) 
holds for all values, q must be constant for all values of a, /9, 7. 

21. Going back to (29), taking Q great enough to allow 
WIQ to be neglected, and simplifying by (46), (43), and (44), we 
find 



and the problem ( 17) of determining /, m, n, subject to (5) 
and (6), to make / s /a + m a /$ + n 3 /7 a maximum or minimum for 
given values of X, n, v, yields the equations 

o>X o>7 4- = ; fOfji ca'm + -^ = ; o>i> to'n + - = ...... (54), 

a p 7 

a>, (a denoting indeterminate multipliers ; whence 

*-=*?+? ....................................... <* 

(1\ / l\a / 1\2 

o>'--) +mU'-gJ +nU' J ...... (56), 



1 - 1 3 



.(57). 



240 LECTURE XV. 

These formulas are not directly convenient for finding I, m, n 
from \, fju, v, cf. 33 (the ordinary formulas for doing so need not be 
written here) ; but they give \, /*, v explicitly in terms of I, m, n 
supposed known ; that is to say, they solve the problem of finding 
the wave-front of the simple laminar wave whose direction of 
vibration is (I, m, n). The velocity is given by 



It is interesting to notice that this depends solely on the direction 
of the line of vibration ; and that (except in special cases, of 
partial or complete isotropy) there is just one wave-front for any 
given line of vibration. These are precisely in every detail the 
conditions of Fresnel's Kinematics of Double Refraction. 

22. Going back to (35) and (36), let us see if we can fit 
them to double refraction with line of vibration in the plane 
of polarization. This would require (36) to be the ordinary ray, 
and therefore requires the fulfilment of (38), as did the other 
supposition ; but instead of (37) we now have [in order to make 
(36) constant] 

A=B = C ........................... (59), 

and therefore each, in virtue of (14), zero ; and 



so that we are driven to complete isotropy. Hence our present 
form ( 7) of the stress-theory of double refraction cannot be fitted 
to give line of vibration in the plane of polarization. We have 
seen ( 21) that it does give line of vibration perpendicular to the 
plane of polarization with exactly Fresnel's form of wave-surface, 
when fitted for the purpose, by the simple assumption that the 
potential energy of the strained solid is expressed by (52) with 
q constant ! It is important to remark that q is the rigidity- 
modulus of the unstrained isotropic solid. 

23. From (58) we see that the velocities of the waves corre- 
sponding to the three cases, 1=1, m= 1, n = 1, respectively are 
V(?/ a )> V(^//5), V(<7/7)- Hence the velocity of any wave whose 
vibrations are parallel to any one of the three principal elongations, 
multiplied by this elongation, is equal to the velocity of a wave in 
the unstrained isotropic solid. 



HYPOTHETICAL ELASTIC SOLID. 241 

[ 24... 34, added March 1899.] Molar 

24. To fix and clear our understanding of the ideal solid, 
introduced in 7 and defined in 20 (52) and in 22, take a bar 
of it of length I when unstrained, and of cross-sectional area A. 
For our present purpose the cross-section may be of any shape, 
provided it is uniform from end to end. Apply opposing forces P 
to the two ends ; and, when there is equilibrium under this stress, 
let the length and cross-section be x and A'. We have A' = lA/x, 
because the solid is incompressible. The proportionate shortenings 
of all diameters of the cross-section will be equal, as there is no 
lateral constraint. Hence if the a, $, 7 of (52) refer to the length 
of our bar and two directions perpendicular to it, we have 

a = |; y9 = 7 = ^ ..................... (60); 

and therefore by (52) 

.................. (61), 



where K denotes the whole work required to make the change 
from I to x in the length of our bar, of which the bulk is I A. 
Hence, as P = dEfdx, we find 

, and SP=^Bx ............ (62). 

Hence, denoting by M the Young's modulus for values of x differ- 
ing not infinitely little from /, and defining it by the formula 



(63). 



Hence augmentation of length from I to x diminishes the Young's 
modulus, as defined above, in the ratio of I- to X s . 

25. The property of our ideal solid expressed by the con- 
stancy of 7, to which we have been forced by the .assumption of 
Fresnel's laws for light traversing a crystal, is interesting in 
respect to the extension of theory and ideas regarding the 
elasticity of solids from infinitesimal strains, for which alone the 
dynamics of the mathematical theory has hitherto been developed, 
to strains not infinitesimal. To contrast the law of relation 
between force and elongation for a rod of our ideal substance, 
as expressed in (<>2) above, and for a piece of indiarubber cord or 
T. L. 16 



242 LECTURE XV. 

Molar. indiarubber band, as shown by experiment, is quite an instructive 
short lesson in the elements of the extended theory. 

26. Ten years ago, because of pressure of other avocations, 
I reluctantly left the stress theory of Frcsuel's laws of double 
refraction without continuing my work far enough to find complete 
expressions for the equilibrium and motion of the elastic solid for 
any infinitesimal deviation whatever from the condition (a, (3, 7). 
Here is the continuation which I felt wanting at that time. 
Let (,T O , y n , z (l ) be the coordinates of a point of the solid in its 
unstrained condition (a = l,/3=l,7 = l); and (x, y, z), the co- 
ordinates of the same point in the strained condition (a, /3, 7). 
If the axes of coordinates are the three lines of maximum, mini- 
max, and minimum elongation, (which are essentially* at right 
angles to one another), we have 



Let the matter at the point (as, y, z) be displaced to (x + %, y + 77, 
z + ) ; , 77, having each, subject only to the condition of no 
change of bulk, any arbitrary value for every point (x, y, z} 
within some finite space >S', outside of which the medium remains 
in the condition (a, /3, 7) undisturbed. 

Let now the variation of the displacement (f, ?;, ) from point 
to point of the solid be so gradual th;it each one of the nine 
ratios 

d d df. drj drj dj,. d d{ d 

dx ' d,y ' dz ' dx ' d>/ ' dz ' dx ' dy' dz " 

is an infinitely small numeric. Consider the system of bodily 
forces, which must be applied to the solid within the disturbed 
region 8, to produce the specified displacement (, 77, ). Let 
Xfl, Ffl, ZQ denote components of the force which must be 
applied to any infinitesimal volume, ft, of matter around the point 
(x, y, z). Our directly solvable problem is to determine X, Y, Z 
for any point, , 77, being given for every point. 

27. Our supposition as to infinitesimals, which is that the 
infinitesimal strain corresponding to the displacement (f, 77, ), is 
superimposed upon the finite strain (a, /3, 7), implies that if instead 

* See Thomson and Tait, Ifi-i. 



FURTHER DEVELOPMENT OF STRESS-THEORY. 243 

of (f, 17, f), the displacement be (c&crj, cf), where c is any numeric Molar, 
less than unity having the same value for every point of the 
disturbed region, the force required to hold the medium in this 
condition is such that instead of X, Y t Z, we have cX, cY, cZ. 
Hence if BE denote the total work required to produce the 
displacement (f, 17, ), we have 

...(66), 



where 1 1 1 dxdydz denotes integration throughout <S'. 
28. Now according to (52) we have 



where a', ft 1 , y denote the squares of the principal elongations 
( 7 above) as altered from a, /9, 7, in virtue of the infinitesimal 
displacement (, 17, ). 

To find a, fi 1 , y, draw, parallel to our primary lines of reference 
OX, OY, OZ, temporary lines of reference through the point 
(* + f. y + n> z + f ) which for brevity we shall call Q. Let P be 
a point of the solid infinitely little distant from Q, and let (f, </, //) 
be its coordinates relatively to these temporary reference lines. 
Relatively to the same lines, let (f , g , /*) be the coordinates of 
the position, P , which P would have if the whole solid were 
unstrained. We have 



**, va-. ,,,vo-.-,, */nr, 

' (68). 



From these we find 

(PQV 



PQ 



A I- + Bin- + CV + 2 (amn + bnl + dm) (69), 



where I, m, n denote * V , -p " , '--, and 



162 



244 



Molar. 



LECTURE XV 

f dt 



5. i 4. rj 

7 I -*- ' 7 

dy \ di 



dz 



...(70). 



dz 



dy) 

The values of a', /3', 7' are the maximum, minimax, and 
minimum values of (09), subject to the condition l 2 + m* + ?? 2 = 1 ; 
and therefore according to the well-known solution of this problem, 
they are the three roots, essentially real, of the cubic 

(A-p)(B-p)(0-p) 

- a- (A-p)- b* (B-p)- c- (C -p) + 2abc = ...... (71). 

Hence, by taking the coefficient of p in the expansion of this, we 
find 

/3V + yV + a/3' = EC +CA+AB- u* -b n --c~ 



where F, G, H are symmetrical algebraic functions of the fourth 
degree of the nine ratios (05). Omitting all terms above the 
second degree, we have 



F - 2 
* 



dy 



- 
dz 



dy 



dz 



__ ...... 

dy dz dz dy 



and symmetrical expressions for G and H. Going back now to 
(67) and remembering that ay = 1 and a'/3'y' = l, and that 
a, /3, y are constant throughout the solid, we find, by (67), 



FURTHER DEVELOPMENT OF STRESS-THEORY. 24.5 

To find III dxdydzF by (73), remark first that by simple in- Molar, 
tegration with respect to y and with respect to z we obtain 

= (75), 



because rj and f each vanish through all the space outside the 
space & For the same reason we find by the well-known integra- 
tion by parts [Lecture X 1 11., equations (7) above] 



We thus have 



2!). Now the condition ot'ft'y = 1 gives 

rff dij d% 

~ + -j + . =0 + terms involving powers, 

and products of the ratios ........ (78). 

Therefore, neglecting higher jM>wers than squares, we get 



<> 

and (77) becomes 



and we have corresponding symmetrical equations for G and H. 
With these used in (74), it becomes 



Taking any one of the nine terms of (81), the third for example, 
and applying to it the process of double integration by parts, we 
find 



246 LECTURE XV. 

Molar. Hence treating similarly all the other terms, we have 



30. This expression for BE must be equal to that of (66) 
above, for every possible value of , ?;, If all values were 
possible, we should therefore have tlie coefficients of , 77, % in (66) 
equal respectively to the coefficients of , 77, in (83). But in 
reality only values of f, 77, are possible which fulfil the condition 
of bulk everywhere unchanged ; and Lagrange's method of inde- 
terminate multipliers adds to the second member of (83), the 
following 



This, treated by the method of double integration by parts, 
becomes 



where or = -X ; s -f , ' + yM (86). 

\dx ay dz / 

31. We must now, according to Lagrange's splendidly 
powerful method, equate separately the coefficients of , 77, 
in (66) to their coefficients in (83) with (85) added to it. Thus 
we find 



a a; y ^ y 2 

as the equations of equilibrium of our solid with every point of 
its boundary fixed, and its interior disturbed from the finitely 
strained condition (a, /3, 7) by forces X, Y, Z, producing infini- 
tesimal displacements (, rj, f) ; subject only to the condition 

^ + * + .0 . ...(88), 

dx dy dz 

to provide against change of bulk in any part. If this equation 
were not fulfilled, the equations (87), with w given by (86), would 
be the solution of a certain definite problem regarding a compres- 
sible homogeneous solid, having certain definite quality of perfect 
elasticity, in respect to changes of shape and bulk, defined by the 
coefficients q and X. From this problem to our actual problem there 



FURTHER DEVELOPMENT OF STRESS-THEORY. 247 

is continuous transition by making X everywhere infinitely great, Molar. 
and therefore the first member of (88), being the dilatation, zero; 
and leaving vr as a quantity to be determined to fulfil the condi- 
tions of any proposed problem. Thus in (87), (88), supposing 
X, Y, Z given, we have four equations for determining the four 
unknown quantities nr, , rj, f. 

32. Suppose now that the solid, after having been in- 
finitesimally disturbed by applied forces from the (a, ft, 7) 
condition, is left to itself. According to D'Alembert's principle, 
the motion is determined by what the equations of equilibrium, 
(87), become with p$, pij, p% substituted for X, Y, Z; p 
denoting the density of the solid. Thus we find for the equations 
of motion 

Vr?.,*. dm rf*if q _., dvr 



33. We are now enabled by these equations to work out 
the problem of wave-motion by a more direct and synthetical 
process than that by which we were led to the solution in 21 
above. The simplest mathematical expression defining plane 
waves in an elastic solid in terms of the notation of (89) is 



p = \x + py + vz 



...(90); 



where X, /*, v are the direction-cosines of the wave-normal, and 
I, in, n those of the line of vibration. Our constraint to incom- 
pressibility gives 

IX + mp. + nv = ..................... (91). 

Eliminating f /, v from (89) by (90), we find 

/ (pv- - qcr 1 ) = X(/y ; nt(ptf - <J0~ 1 ) = pqu ', n (pv 2 qy~ l ) = vqw 

..................... (92), 

where 



q p p 2 
These equations agree with (54) if for pv 2 we take qa'. 



248 LECTURE XV. 

Molar. 34. Multiplying equations (92) by 

\'(pv- qa~ l ). pKpv 3 f?/?" 1 ), i' (pi' 2 qy~ l ) 
respectively, adding, and using (91), we find 

+ ^ + "" =0 (94). 

pv 2 qor 1 pv- qp l pv qy~ l 

This is a quadratic for the determination of <-, with its two roots 
essentially real and positive. Again, by the equation 

we find from (92) 



Equations (94) and (95) give us jr and o> ; and (92) now gives 
explicitly /, in, n, when X, p, i> are given. Thus we complete the 
determination of (I, in, n), the direction of vibration, in terms of 
(\, p. i'), the wave-normal, and the constant coefficients 
(/a- 1 , q3~ l , qy~ l . 

Our solution is identical with Fresnel's, and implies exactly the 
same shape of wave-surface. 

[ 35... 47, added April 1901.] 

35. It will be convenient henceforth to take a, b. C instead 
of or-'. H, y~-. Thus, according to 7 (1), if .r,,, y v , r and 
.r, i/. - denote respectively the coordinates of one and the same 
particle in the unstrained and in the strained solid, we have*, 



*=- 



or a ' : /= -y- : <j = -** ...(97): 

where t? = '- ; /= ; <J = ~ " (98). 

From (97) we get 

a= rr e ; b = i!/ : C= IT^ 

whence (1 + ^)^1+^)^1^,7)^=1 (100). 

* It would have been better from the beginning to have taken single letters 
instead ofa~^, 8~^ . -y~i. 



FURTHER DEVELOPMENT OF STRESS-THEORY. 249 

The principal elongations to pjvss from the unstrained to the Molar. 
strained solid are B" 1 , b" 1 , t~ l ; and the principal ratios of elonga- 
tion from the strained to the unstrained solid are a, b, C. And if 
E denote the work per unit volume required to bring the solid 
from the unstrained to the strained condition, we have by (52) 
=^y(a' + b a + C 2 -3) ..................... (101). 

From this, remembering that abc = 1, we find 

P = -(qtf + *r); Q = -(qfr+*r), = - (tfC* + tsr) ....(102) ; 

2 + e 




where P, Q, R denote the normal components of force per unit area 
(pulling outward when ]>ositiv f e) on the three pairs of faces of a 
rectangular parallelepiped required to keep it in the state of strain 
a, b, C, with principal elongations perpendicular to the pairs of 
faces ; and vr denotes an arbitrary pressure uniform in all direc- 
tions. The proof is as follows : Consider a cube of unit edges in 
the unstrained solid. In the strained condition the lengths of its 
edges are I/a, 1/b, 1/C, and the areas of its faces are fl, b, C. 
Hence the equation of work done to augmentation of energy pro- 
duced in changing n, b, C, to a + Sfl, b -f 8b, C + Bt, is 

PaS i -I- QbS g + cS i = q&E=q(aBa + bSb + cSc) ...... (104) ; 

and by flbt = 1 (constancy of volume) we have 



Hence by Lagrange's method 

--(-!* 



+ Sc ...... (106), 

where v denotes an "indeterminate multiplier"; and we have 
definitively 

-P=ga 3 + m; - Q = q\3r + v ; -R = q? + m ....... (107), 



250 LECTURE XV. 

which prove (102). The meaning of -OT here, an arbitrary magni- 
tude, is a pressure uniform in all directions, which, as the solid 
is incompressible, may be arbitrarily applied to the boundary of 
any portion of it without altering any of the effective conditions. 

36. By 33 (92), we see that the propagational velocities 
of waves whose lines of vibration are parallel to OX, OY, OZ are 
respectively 



Thus the propagational velocity of a wave whose front is parallel 
to the plane YOZ (\ = l, /u, = 0, y = 0) is fa A/- if< its linc of 



vibration is parallel to OY (1 = 0, tit}, n = 0), and is C A / 



if its line of vibration is parallel to OZ (I 0, iu = Q, n = 1) ; 
ami similarly in respect to waves whose fronts are parallel to 
ZX and XY. 

For brevity we shall call a A /- , l) A /~, C A /- the prin- 

V P V p V p 

cipal velocities of light in the crystal, and OX, OY, OZ its three 
principal lines of symmetry. We are precluded from calling 
these lines optic axes, by the ordinary usage of the word axis in 
respect to uni-axial and bi-axial crystals. 

37. To help us to thoroughly understand the dynamics of 
the stress theory of double refraction, consider as an example 
aragonite, a bi-axial crystal of which the three principal refractive 
indices are 1-5301, I'GSIG, l'G859. If, as according to our stress- 
theory, optic ujolotropy is due to unequal extension and con- 
traction in different directions of the ether within the crystal 
with volume unchanged, the principal elongations being in simple 
proportion to the three principal velocities of light within it, 
annulment of the extension and contraction would give isotropy 
with refractive index l'G'309, being the cube root of the product 
of the three principal indices. Hence, if we call V the pro- 
pagational velocity corresponding to this mean index, the three 
principal velocities in the actual crystal (being inversely as the 
refractive indices) are l'OG59F, "9G87 V, '9G79F, of which the 



FURTHER DEVELOPMENT OF STRESS-THEORY. 



251 



product is K J . Hence according to our notation of 7 and 35 Molar, 
above we have 

a = 10659; b = '9687; = '9679. 

38. Let the dotted ellipse in the diagram represent a' cross 
section of an elliptic cylindric portion of aragonite having its 
axis of figure in the direction of maximum elongation of the 
ether. Let the diameter A' A be the direction of maximum 




FIG. 1. 

contraction, and B'B that of minimax (elongation-contraction), 
being in fact in this case a line along which there is elongation. 
The circle in the diagram shows the undisturbed positions of the 
particles of ether, which in the crystal are forced to the positions 
shown by the dotted ellipse. The axes of the ellipse are equal 



252 LECTURE XV. 

respectively to 1/1-0659 and 1/-9G87 of the diameter of the 
circle. 

First, consider waves whose fronts are parallel to the plane 
of the diagram. If their vibrations are parallel to OA, the 
direction of maximum contraction, their velocities of propagation 
are l'OG59 V. If their vibrations are parallel to OB their 
velocities are '9687 V. 

Secondly, consider waves whose fronts are perpendicular to 
the plane of the diagram. The propagational velocity of all of 
them whose vibrations are perpendicular to this plane is '9079 V, 
their lines of vibration being all in the direction of maximum 
elongation. If their vibrations are in the plane of the diagram 
and parallel to OA, their velocity is l'OG59 V. If their vibrations 
are parallel to OB, their velocities are '9687 V. The former of 
these is the greatest and the latter the least of the propagational 
velocities of all the waves whose vibrations are in this plane : the 
ratio of the one to the other is TIOO. 

30. It is interesting, on looking at our diagram, to think 
how slight is the distortion of the ether required to produce the 
double refraction of aragonite. If the diagram bad been made 
{'or the plane ZOX containing the lines of vibration for greatest 
and least velocities (ratio of greatest to least I'lOl), the increase 
of ellipticity of the ellipse would not have been perceptible to the 
eye. Only about one and a half per cent greater ratio of the 
difference of the diameters of the ellipse to the diameter of the 
circle would be shown in the corresponding diagram for Iceland 
spar (ratio of greatest to least refractivity = I'llo). For nitrate 
of soda the ratio is somewhat greater still (1-188). For all other 
crystals, so far as I know, of which the double refraction has been 
measured it is less than that of Iceland spar. With such slight 
distortions of ether within the crystal as the theory indicates for 
all these known cases, we could scarcely avoid the constancy of 
our coefficient q, to the assumption of which we were forced, 
20, 25 above, in order to fit the theory to Fresnel's laws for 
light traversing a crystal, irrespectively of smallness or greatness 
of the amount of double refraction. Hence we see that, in fact, 
without any arbitrary assumption of a new property or new pro- 
perties of ether, we have arrived at what would be a perfect 
explanation of the main phenomena of double refraction, if we 



FURTHER DEVELOPMENT OF STRESS-THEORY. 253 

could but see how the molecules of matter could so act upon Molecc 
ether as to give a stress capable of producing the strain with 
which hitherto we have been dealing. Inability to see this has 
prevented me, and still prevents me, from being convinced that 
the stress-theory gives the true explanation of double refraction. 

40. Now, quite recently, it has occurred to me that the 
difficulty might possibly be overcome if, as seems to me necessary 
on other grounds, we adopt a hypothesis regarding the motion of 
ponderable matter through ether, which I suggested a year ago in 
a Friday Evening Lecture, April 27th, 1900, to the Royal Insti- 
tution (reproduced in the present volume as Appendix B) and 
with somewhat full detail in a communication* of last July to the 
Royal Society of Edinburgh, and to the Congres Internationale 
de Physique f in Paris last August (Appendix A, below). Accord- 
ing to this hypothesis ether is a structureless continuous elastic 
solid j>ervading all space, and occupying space jointly with the 
atoms of ponderable matter wherever ponderable matter exists ; 
and the action between ponderable matter and ether consists of 
attractions and repulsions throughout the volume of space occupied 
by each atom. These attractions and repulsions would be essen- 
tially ineffective if ether were infinitely resistant against forces 
tending to condense or dilate it, that is to say, if ether were 
absolutely incompressible. Hence, while acknowledging that ether 
resists forces tending to condense it or to dilate it, sufficiently to 
account for light and radiant heat by waves of purely transverse 
vibration (eqiii-voluminal waves as I have called them), it must, 
by contraction or dilatation of bulk, yield to compressing or 
dilating forces sufficiently to account for known facts dependent 
on mutual forces between ether and ponderable matter. I have 
suggested* that there may be oppositely electric atoms which 
have the properties respectively of condensing and rarefying the 
ether within them. But for the present, to simplify our sup- 
positions to the utmost, I shall assume the law of force between 
the atom and the ether within it to be such that the average 
density of the ether within the atom is equal to the density of 

Proc. Roy. Soc. Edin. July, 1900 ; P*i7. Mag. Ang. 1900. 
t Report*, Vol. 11. page 1. 

* Congrts Internationale de Phytiquf, Reportt, Vol. II. p. 19 ; also Phil. Mag. 
Sept. 1900. 



254 



LECTURE XV. 



[olecular. the undisturbed ether outside, and that concentric spherical sur- 
faces within the atom are surfaces of equal density. The forces 
between ether and atoms we can easily believe to be enormous in 
comparison with those called into play outside the atoms, in virtue 
of undulatory or other motion of ether and elasticity of ether; 
as for instance in interstices between atoms in a solid body, or 
in the space traversed by the molecules of a gas according to 
the kinetic theory of gases, or in the vacuum attainable in our 
laboratories, or in interstellar space. 

41. To fix our ideas let ether experience condensation in 
the central part of the atom and rarefaction in the outer part 
according to the law explained generally in the first part of 5, 
Appendix A, and represented particularly by the formulas (9) and 
(11) of that section, and fully described with numerical and 
graphical illustrations for a particular case in 5 8. Looking 




Fio. 2. 



to cols. 3 and 4 of Table I, Appendix A, we see that, at distance 
r = '50 from the centre of the atom, the density of the ether is 
equal to the undisturbed density outside the atom ; and that from 
r = '50 to r = 1 the density decreases to a minimum, '35, at 
r= '865, and augments thence to the undisturbed density, 1, at 
the boundary of the atom (r = l). In each of the two atoms 
represented in fig. 2, the spherical surface of undisturbed density 
is indicated by a dotted circle, that of minimum density by a fine 
circle, and the boundary of the atom by a heavy circle. Because 
the ethereal density decreases uninterruptedly from the centre to 



FURTHER DEVELOPMENT OF STRESS-THEORY. 255 



the surface of minimum density, the force exerted byt 
the ether must be towards the centre throughflTTfrthis spherical 
space ; and because the ethereal densityJifcreases uninterruptedly 
outwards from the surface of mininwfn density to the boundary of 
the atom, the force exerted by the atom on the ether must be 
repulsive in every part of the shell outside that surface. 

42. Suppose now two atoms to be somehow held together in 
some such position as that represented in fig. 2, overlapping one 
another throughout a lens-shapod space lying outside the *urm63 
of minimum ethereal density in each atom. The rarefaction of 
the ether in this lens-shaped space is, by the combined action of 
the two atoms, greater than at equal distances from the centre in 
non-overlapping portions of both the atoms. Hence, remembering 
that each atom while attracting the ether in its central parts 
repels the ether in every part of it outside the spherical surface of 
minimum density, we see that the repulsion of each atom on the 
ether in the lens-shaped volume of overlap is less than its repulsion 
in the contrary direction on an equal and similar portion of the 
ether within it on the other side of its centre. Hence the re- 
actions of the ether on the atoms are forces tending to bring them 
together; that is to say apparent attractions. These apparent 
attractions are balanced by repulsions between the atoms them- 
selves if two atoms rest stably as indicated in fig. 2. It seems not 
improbable that these are the forces concerned in the equilibrium 
of the two atoms of the known diatomic simple gases N 2 , O 2 , H,. 
I assume that the law of force between ether and atoms and the 
law of elasticity of the ether under this force to be such that no 
part of the ether outside the atoms experiences any displacement 
in consequence of the displacements actually produced within the 
space occupied separately and jointly by the two atoms. This 
implies that the ether drawn away from the lenticular space of 
overlap by the extra rarefaction there, is taken in to the central 
regions of the two atoms in virtue of the attractions of the atoms 
on the ether in those regions. In an addition to Lecture XVIII. 
on the reflection and refraction of light, I shall have occasion to 
give explanation and justification of this assumption. 

43. As a representation of an optically isotropic crystal, 
consider a homogeneous assemblage of atoms for simplicity taken 



ZOO LECTURE XV. 

Molecular, in cubic order as shown in fig. 3. Each one of the outermost 
atoms experiences resultant force inwards from the ether within 
it, and this force is balanced by repulsion exerted upon it by the 
atom next it inside. For every atom except those lying in the 
outer faces, the forces which it experiences in different directions 
from the ether within it balance one another : and so do the forces 
which it experiences from the atoms around it. But each of the 
outermost atoms experiences a resultant repulsion from the other 




Fio. 3. 

atoms in contact with it, and this repulsion outwards is balanced 
by a contrary attraction of the ether on the atom. Hence the 
outermost atoms all round a cube of the crystal exert an outward 
pull upon the ether within the containing cube. In accordance 
with the assumption stated at the end of 42, the position and 
shape of every particle of ether outside the atoms is undisturbed 
by the forces exerted by the atoms in the spaces occupied by them 
separately and jointly ; and it is only in these spaces that the 
ether is disturbed by the action of the atoms. 

44. Suppose now that by forces applied to the atoms as 
indicated by the arrow-heads in fig. 4, the distances between the 
centres of contiguous atoms arc increased and diminished as shown 
in the diagram. With this configuration of the atoms the ether 
is pulled outwards by the atoms with stronger forces in the 
direction parallel to AC, BD than in directions parallel to AB 
and CD. Hence as ( 42, 43 above) the ether is unstressed and 
unstrained in the interstices between the atoms, the ether within 
the atoms experiences an excess of outward pulling stress in the 



FURTHER DEVELOPMENT OF STRESS-THEORY. 257 

direction parallel to AC and BD above outward pulling stress in Molecula 
the direction perpendicular to these lines : and therefore, if there 
ivere no ivolotropy of inertia, and if the stress theory which we 
have worked out ( 35, 36) for homogeneous ether is applicable 



! 1 J 




FIG. 4. 

to average action of the whole ether with its great inequalities of 
density in the space occupied by the assemblage of atoms, the 
propagational velocity of distortional waves would be greater when 
the direction of vibration is parallel to AC and BD than when it 
is parallel to AB and CD. But alas! this is exactly the reverse 
of what, thirteen years ago, Kerr, by experimental research of a 
rigorously testing character, found for the bi-refringent action of 
strained glass*, the existence of which had been discovered by 
Sir David Brewster seventy years previously. We are forced to 
admit that one or both of our two " if's must be denied. 

45. It seems to me now worthy of consideration whether 
the true explanation of the double refraction of a natural crystal 
or of a piece of strained glass may possibly be that given by 
Glazebrook in another paperf in the same volume of the Philo- 
sophical Magazine as Kerr's already referred to. Glazebrook made 

* " Experiments on the Birefringent Action of Strained Glass," PMl. Mag. Oct 
1888, p. 339. 

t "On the Application of Sir William Thomson's Theory of a Contractile 
Ether to Double Refraction, Dispersion, Metallic Reflexion, and other Optical 
Problems." Phil. Mag. Dec. 1888, p. 524. 

T. L. 17 



258 LECTURE XV. 

Molar, the remarkable discovery expressed in his equation (14) (p. 525) 
that, when the propagational velocity of the condensational- 
rarefactional wave is zero, the two propagational velocities due to 
seolotropic inertia in a distortional wave with same wave-front* 
are the same as those given by my equation (94), and originally by 
Fresnel; from which it follows that the wave-surface is exactly 
Fresnel's. It is certainly a most interesting result that the wave 
surface should be exactly Fresnel's, whether the optic *olotropy is 
due to difference of stress in different directions in an incompres- 
sible elastic solid, or to seolotrupy of inertia in an ideal elastic 
solid endowed with a negative compressibility modulus of just 
such value as to make the velocity of the condensational-rare- 
factional wave zero. In Glazebrook's theory the direction of 
vibration is perpendicular to the line of the ray, and in the 
vibratory motion the solid experiences a slight degree of change 
of bulk combined with pure distortion. In the stress theory of 
7... 34 the line of vibration is exactly in the wave-front or 
perpendicular to the wave normal and there is no change of bulk. 

46. In Lecture XVIII, when we are occupied with the 
reflection and refraction of light, we shall see that Fresnel's 
formula for the three rays, incident, reflected, and refracted, when 
the line of vibration is in their plane, is a strict dynamical conse- 
quence of the assumption of zero velocity for the condensational- 
rarefactional wave in both the mediums, or in one of the mediums 
only while the other medium is incompressible. But the difficulties 
of accepting zero velocity for condensational-rarefactional wave, 
whether in undisturbed ether through space or among the atoms 
of ponderable matter, arc not overcome. 

47. It will be seen, ( 3, 4, 7, 21, 22, 24, 25, 33, 35, 37, 38, 
39, 40, 45 above,) that after earnest and hopeful consideration of 
the stress theory of double refraction during fourteen years, 
I am unable to see how it can give the true explanation 
either of the double refraction of natural crystals, or of double 
refraction induced in isotropic solids by the application of unequal 
pressures in different directions. Nevertheless the mathematical 
investigations of 7... 21 and 24... 34, interesting as they are 

* I, m, n are the direction-cosines of the normal to the wave- front in Glaze- 
brook's paper, and X, /*, v, in my 18, 21, 33, 34 above, 



FURTHER DEVELOPMENT OF STRESS-THEORY. 259 

in the abstract dynamics of a homogeneous incompressible elastic Molar, 
solid, have an important application in respect to the influence of 
ponderable matter on ether. They prove in fact the truth of the 
assumption at the end of 42, however the forces within the atoms 
are to be explained ; because any distortion of ether in the space 
around a crystal, even so slight as that illustrated in fig. 1, 38 
above, would produce double refraction in air or vacuum outside 
the crystal, not quite as intense outside as inside ; not vastly less, 
close to the outside ; and diminishing with distance outside but 
still quite perceptible, to distances of several diameters of the 
ma. 



172 



LECTURE XVI. 

WEDNESDAY, October 15, 5 P.M. 

[THIS was a double lecture ; but as the substance of the first 
part, with amplification partly founded on experimental discoveries 
by many workers since it was delivered, has been already repro- 
duced in dated additions on pp. 148157 and 176 184 above, 
only the second part is here given.] 

I want now to go somewhat into detail as to absolute 
magnitudes of masses and energies, in order that there may be 
nothing indefinite in our ideas upon this part of our subject ; and 
I commence by reading and commenting on an old article of mine 
relating to the energy of sunlight and the density of ether. 

[Nov. 20, 1899... March 28, 1901. From now, henceforth till 
the end of the Lectures, sections will be numbered continuously.] 

NOTE ON THE POSSIBLE DENSITY OF THE LUMINIFEROUS 
MEDIUM AND ON THE MECHANICAL VALUE OF A CUBIC 
MiLE*f OF SUNLIGHT. 

[From Edin. Royal Soc. Trans., Vol. xxi. Part i. May, 1854 ; Phil. Mag. 
ix. 1854 ; Comptes Rendus, xxxix. Sept. 1854 ; Art. LXVII. of Math, and 
Phys. Paper sJ] 

Molar. 1. That there must be a medium forming a continuous 
material communication throughout space to the remotest visible 
body is a fundamental assumption in the undulatory Theory 
of Light. Whether or not this medium is (as appears^ to 

* [Note of Dec. 22, 1892. The brain-wasting perversity of the insular inertia 
which still condemns British Engineers to reckonings of miles and yards and feet 
and inches and grains and pounds and ounces and acres is curiously illustrated by 
the title and numerical results of this Article as originally published.] 

t [Oct. 13, 1899. In the present reproduction, as part of my Lee. XVI. of 
Baltimore, 1884, I suggest cubic kilometre instead of " cubic mile " in the title and 
use the French metrical system exclusively in the article.] 

J [Oct. 13, 1899. Not so now. I did not in 1854 know the kinetic theory of gases.] 



UNDISTURBED DENSITY OP HOMOGENEOUS ETHER. 261 

me most probable) a continuation of our own atmosphere, Molar, 
its existence is a fact that cannot be questioned, when the 
overwhelming evidence in favour of the undulatory theory is 
considered; and the investigation of its properties in every 
possible way becomes an object of the greatest interest. A first 
question would naturally occur, What is the absolute density of 
the luminiferous ether in any part of space ? I am not aware of 
any attempt having hitherto been made to answer this question, 
and the present state of science does not in fact afford sufficient 
data. It has, however, occurred to me that we may assign an 
inferior limit to the density of the luminiferous medium in inter- 
planetary space by considering the mechanical value of sunlight 
as deduced in preceding communications to the Royal Society 
[Tram. R. S. E.; Mechanical Energies of the Solar System; re- 
published as Art. LXVI. of Math, and Phys. Papers] from Pouillet's 
data on solar radiation, and Joule's mechanical equivalent of 
the thermal unit. Thus the value of solar radiation per second 
per square centimetre at the earth's distance from the sun, 
estimated at 1235 cm. -grams, is the same as the mechanical 
value of sunlight in the luminiferous medium through a space 
of as many cubic centimetres as the number of linear centimetres 
of propagation of light per second. Hence the mechanical value 
of the whole energy, kinetic and potential, of the disturbance 
kept up in the space of a cubic centimetre at the earth's distance 

, . 1235 412 , 
from the sun*, is - TT^,, or ^ of a cm.-gram. 

2. The mechanical value of a cubic kilometre of sunlight is 
consequently 412 metre-kilograms, equivalent to the work of one 
horse-power for 5*4 seconds. This result may give some idea of 
the actual amount of mechanical energy of the luminiferous 
motions and forces within our own atmosphere. Merely to com- 
mence the illumination of eleven cubic kilometres, requires an 
amount of work equal to that of a horse-power for a minute; 
the same amount of energy exists in that space as long as 
light continues to traverse it; and, if the source of light be 
* The mechanical value of sunlight in any space near the sun's surface must 
be greater than in an equal space at the earth's distance, in the ratio of the square 
of the earth's distance to the square of the sun's radius, that is, in the ratio of 
46,000 to 1 nearly. The mechanical value of a cubic centimetre of sunlight near 

the sun must, therefore, be or about -0019 of a cm.- 



262 LECTURE XVI. 

Molecular, suddenly stopped, must pass from it before the illumination 
ceases*. The matter which possesses this energy is the lumini- 
ferous medium. If, then, we knew the velocities of the vibratory 
motions, we might ascertain the density of the luminiferous 
medium ; or, conversely, if we knew the density of the medium, 
we might determine the average velocity of the moving particles. 

3. Without any such definite knowledge, we may assign a 
superior limit to the velocities, and deduce an inferior limit to the 
quantity of matter, by considering the nature of the motions 
which constitute waves of light. For it appears certain that the 
amplitudes of the vibrations constituting radiant heat and light 
must be but small fractions of the wave-lengths, and that the 
greatest velocities of the vibrating particles must be very small 
in comparison with the velocity of propagation of the waves. 

4. Let us consider, for instance, homogeneous plane polarized 
light, and let the greatest velocity of vibration be denoted by v ; 
the distance to which a particle vibrates on each side of its 
position of equilibrium, by A; and the wave-length, by X. Then, 
if V denote the velocity of propagation of light or radiant heat, 
we have 

- - 27T - ' 

V~ X' 

and therefore if A be a small fraction of X, v must also be a small 
fraction (2?r times as great) of V. The same relation holds for 
circularly polarized light, since in the time during which a particle 
revolves once round in a circle of radius .A, the wave has been pro- 
pagated over a space equal to X. Now the whole mechanical value 
of homogeneous plane polarized light in an infinitely small space 
containing only particles sensibly in the same phase of vibration, 
which consists entirely of potential energy at the instants when 
the particles are at rest at the extremities of their excursions, 
partly of potential and partly of kinetic energy when they are 
moving to or from their positions of equilibrium, and wholly of 
kinetic energy when they are passing through these positions, is 
of constant amount, and must therefore be at every instant equal 
to half the mass multiplied by the square of the velocity which the 
particles have in the last-mentioned case. But the velocity of any 
* Similarly we find 4140 horse-power for a minute as the amount of work 
required to generate the energy existing in a cubic kilometre of light near the sun. 



KINETIC AND POTENTIAL ENERGY OF LIGHT. 263 

particle passing through its position of equilibrium is the greatest Molecular, 
velocity of vibration. This we have denoted by v; and, there- 
fore, if p denote the quantity of vibrating matter contained in a 
certain space, a space of unit volume for instance, the whole me- 
chanical value of all the energy, both kinetic and potential, of the 
disturbance within that space at any time is pv*. The mechani- 
cal energy of circularly polarized light at every instant is (as has 
been pointed out to me by Professor Stokes) half kinetic energy 
of the revolving particles and half potential energy of the dis- 
tortion kept up in the luminiferous medium ; and, therefore, v 
being now taken to denote the constant velocity of motion of each 
particle, double the preceding expression gives the mechanical 
value of the whole disturbance in a unit of volume in the present 



5. Hence it is clear, that for any elliptically polarized light 
the mechanical value of the disturbance in a unit of volume will 
be between $pv* and pv 1 , if v still denote the greatest velocity of 
the vibrating particles. The mechanical value of the disturbance 
kept up by a number of coexisting series of waves of different 
periods, polarized in the same plane, is the sum of the mechanical 
values due to each homogeneous series separately, and the greatest 
velocity that can possibly be acquired by any vibrating particle is 
the sum of the separate velocities due to the different series. 
Exactly the same remark applies to coexistent series of circularly 
polarized waves of different periods. Hence the mechanical value 
is certainly less than half the mass multiplied into the square of 
the greatest velocity acquired by a particle, when the disturbance 
consists in the superposition of different series of plane polarized 
waves ; and we may conclude, for every kind of radiation of light 
or heat except a series of homogeneous circularly polarized waves, 
that the mechanical value of the disturbance kept up in any space 
is less than the product of the mass into the square of the greatest 
velocity acquired by a vibrating particle in the varying phases of its 
motion. How much less in such a complex radiation as that of 
sunlight and heat we cannot tell, because we do not know how 
much the velocity of a particle may mount up, perhaps even to 
a considerable value in comparison with the velocity of propaga- 
tion, at some instant by the superposition of different motions 
chancing to agree ; but we may be sure that the product of the 



264 LECTURE XVI. 

Molecular, mass into the square of an ordinary maximum velocity, or of the 
mean of a great many successive maximum velocities of a vibrat- 
ing particle, cannot exceed in any great ratio the true mechanical 
value of the disturbance. 

6. Recurring, however, to the definite expression for the 
mechanical value of the disturbance in the case of homogeneous 
circularly polarized light, the only case in which the velocities 
of all particles are constant and the same, we may define the 
mean velocity of vibration in any case as such a velocity that 
the product of its square into the mass of the vibrating par- 
ticles is equal to the whole mechanical value, in kinetic and 
potential energy, of the disturbance in a certain space traversed 
by it ; and from all we know of the mechanical theory of undula- 
tions, it seems certain that this velocity must be a very small 
fraction of the velocity of propagation in the most intense light 
or radiant heat which is propagated according to known laws. 
Denoting this velocity for the case of sunlight at the earth's 
distance from the sun by v, and calling W the mass in grammes of 
any volume of the luminiferous ether, we have for the mechanical 
value of the disturbance in the same space, in terms of terrestrial 
gravitation units, 

W 

v-, 
9 

where g is the number 981, measuring in (c.G.s.) absolute units 
of force, the force of gravity on a gramme. Now, from Pouillet's 
observation, we found in the last footnote on 1 above, 

1235 x 46000 , 

pr - tor the mechanical value, in centimetre-grams, 

of a cubic centimetre of sunlight in the neighbourhood of the 
sun; and therefore the mass, in grammes, of a cubic centimetre 
of the ether, must be given by the equation, 

W - ??J_ X -1? 35 x 4600 

v-V 

If we assume v = - V. this becomes 
n 

w 981 x 1235 x 46000 981 x 1235 x 46000 

-w- *=-~^i&r *,-> 

20-64 



RIGIDITY OF ETHER. 265 

and for the mass, in grammes, of a cubic kilometre we have Molecular. 

20-64 
-LOT*"' 

7. It is quite impossible to fix a definite limit to the ratio 
which v may bear to V\ but it appears improbable that it could 
be more, for instance, than -fa, for any kind of light following the 
observed laws. We may conclude that probably a cubic centimetre 
of the luminiferous medium in the space near the sun contains 
not less than 516 x 10"* of a gramme of matter; and a cubic 
kilometre not less than 516 x 10~ 5 of a gramme. 

8. [Nov. 16, 1899. We have strong reason to believe that 
the density of ether is constant throughout interplanetary and 
interstellar space. Hence, taking the density of water as unity 
according to the convenient French metrical system, the preceding 
statements are equivalent to saying that the density of ether in 
vacuum or space devoid of ponderable matter is everywhere 
probably not less than 5 x 10~ 18 . 

Hence the rigidity, (being equal to the density multiplied 
by the square of the velocity of light), must be not less than 
4500 dynes* per square centimetre. With this enormous value 
as an inferior limit to the rigidity of the ether, we shall see in an 
addition to Lecture XIX. that it is impossible to arrange for a 
radiant molecule moving through ether and displacing ether by 
its translatory as well as by its vibratory motions, consistently 
with any probable suppositions as to magnitudes of molecules 
and ruptural rigidity-modulus of ether; and that it is also 
impossible to explain the known smallness of ethereal resistance 
against the motions of planets and comets, or of smaller ponder- 
able bodies, such as those we can handle and experiment upon in 
our abode on the earth's surface, if the ether must be pushed 
aside to make way for the body moving through it. We shall find 
ourselves forced to consider the necessity of some hypothesis for 
the free motion of ponderable bodies through ether, disturbing it 
only by condensations and rarefactions, with no incompatibility 
in respect to joint occupation of the same space by the two 
substances.] See Phil. Mag, Aug. 1900, pp. 181198. 

* See Math, and Phyi. Paper*, Vol. in. p. 522 ; and in last line of Table 4, 
for " / 10-* 2 " substitute " /> < lO'* 1 ." 



266 LECTURE XVI. 

Molar. 9. I wish to make a short calculation to show how much com- 
pressing force is exerted upon the luminiferous ether by the sun's 
attraction. We are accustomed to call ether imponderable. How 
do we know it is imponderable ? If we had never dealt with air 
except by our senses, air would be imponderable to us; but we 
know by experiment that a vacuous glass globe shows an increase 
of weight when air is allowed to now into it. We have not the 
slightest reason to believe the luminiferous ether to be imponder- 
able. [Nov. 17, 1899. I now see that we have the strongest 
possible reason to believe that ether is imponderable.] It is just 
as likely to be attracted to the sun as air is. At all events the 
onus of proof rests with those who assert that it is imponderable. 
I think we shall have to modify our ideas of what gravitation is, 
if we have a mass spreading through space with mutual gravita- 
tions between its parts without being attracted by other bodies. 
[Nov. 17, 1899. But is there any gravitational attraction between 
different portions of ether ? Certainly not, unless either it is 
infinitely resistant against condensation, or there is only a finite 
volume of space occupied by it. Suppose that ether is given uni- 
formly spread through space to infinite distances in all directions. 
Any large enough spherical portion of it, if held with its surface 
absolutely fixed, would by the mutual gravitation of its parts become 
heterogeneous ; and this tendency could certainly not be counter- 
acted by doing away with the supposed rigidity of its boundary 
and by the attraction of ether extending to infinity outside it. 
The pressure at the centre of a spherical portion of homogeneous 
gravitational matter is proportional to the square of the radius, 
and therefore, by taking the globe large enough, may be made as 
large as we please, whatever be the density. In fact, if there 
were mutual gravitation between its parts, homogeneous ether 
extending through all space would be essentially unstable, unless 
infinitely resistant against compressing or dilating forces. If we 
admit that ether is to some degree condensible and extensible, 
and believe that it extends through all space, then we must 
conclude that there is no mutual gravitation between its parts, 
and cannot believe that it is gravitationally attracted by the sun 
or the earth or any ponderable matter; that is to say, we must 
believe ether to be a substance outside the law of universal 
gravitation.] 



WEIOHT OF ETHER, IF GRAVITATIONAL 267 

10. In the meantime, it is an interesting and definite question Molar. 
to think of what the weight of a column of luminiferous ether of 
infinite height resting on the sun, would be, supposing the sun 
cold and quiet, and supposing for the moment ether to be 
gravitationally attracted by the sun as if it were ponderable 
matter of density 5 x 10~". You all know the theorem for mean 
gravity due to attraction inversely as the square of the distance 
from a point. It shows that the heaviness of a uniform vertical 
column AB, of mass w per unit length, and having its length in a 
line through the centre of force C, is 

mw mw mw . f ~ n 



where m denotes the attraction on unit of mass at unit distance. 
Hence writing for mw/CA, mwCA/CA\ we see that the attraction 
on an infinite column under the influence of a force decreasing 
according to inverse square of distance, is equal to the attraction 
on a column equal in length to the distance of its near end from 
the centre, and attracted by a uniform force equal to that of 
gravity on the near end. The sun's radius is 697 x 10" cms. and 
gravity at his surface is 27 times* terrestrial gravity, or say 
27000 dynes per gramme of mass. Hence the sun's attraction on 
a column of ether of a square centimetre section, if of density 
5 x 10~ >8 and extending from his surface to infinity, would be 
9*4 x 10~ s of a dyne, if ether were ponderable. 

11. Considerations similar to those of November 1899 in- 
serted in 9 above lead to decisive proof that the mean density of 
ponderable matter through any very large spherical volume of 
space is smaller, the greater the radius ; and is infinitely small for 
an infinitely great radius. If it were not so a majority of the 
bodies in the universe would each experience infinitely great 
gravitational force. This is a short statement of the essence of 
the following demonstration. 

12. Let V be any volume of space bounded by a closed 
surface, S, outside of which and within which there are ponderable 
bodies ; M the sum of the masses of all these bodies within V ; 

* This is founded on the following values for the sun's mass and radius and the 
earth's radius : sun's mass = S24000 earth's mass; sun's radius = 697000 kilo- 
metres; earth's radius =6871 kilometres. 



268 LECTURE XVI. 

Molar, and p the mean density of the whole matter in the volume V. 
We have 

M = P V .............................. (1). 

Let Q denote the mean value of the normal component of the 
gravitational force at all points of S. We have 

QS = lirM = 7rpV ..................... (2), 

by a general theorem discovered by Green seventy-three years 
ago regarding force at a surface of any shape, due to matter 
(gravitational, or ideal electric, or ideal magnetic) acting according 
to the Newtonian law of the inverse square of the distance. It 
is interesting to remark, that the surface-integral of the normal 
component force due to matter outside any closed surface is zero 
for the whole surface. If normal component force acting inwards 
is reckoned positive, force outwards must of course be reckoned 
negative. In equation (2) the normal component force may be 
outwards at some points of the surface S, if in some places the 
tangent plane is cut by the surface. But if the surface is 
wholly convex, the normal component force must be everywhere 
inwards. 

18. Let now the surface be spherical of radius r. We have 
= 47r^; F = 4 3 77 >; V = ^rS ............... (3). 

Hence, for a spherical surface, (2) gives 

4-7T M 



This shows that the average normal component force over the 
surface S is infinitely great, if p is finite and r is infinitely great, 
which suffices to prove 11. 

14. For example, let 

r= 150. 10". 200 . 10" = 3-09 . 10 16 kilometres ...... (5). 

This is the distance at which a star must be to have parallax one 
one-thousandth of a second ; because the mean distance of the 
earth from the sun is one-hundred-and-fifty-million kilometres, and 
there are two-hundred-and-six-thousand seconds of angle in the 
radian. Let us try whether there can be as much matter as a 



GRAVITATIONAL MATTER WITHIN VERY LARGE VOLUME. 2^9 

thousand-million times the sun's mass, or, as we shall say foif Molar, 
brevity, a thousand-million suns, within a spherical surface of that 
radius (5). The sun's mass is 324,000 times the earth's mass; and 
therefore our quantity of matter on trial is 3*24 . 10" times the 
earth's mass. Hence if we denote by g terrestrial gravity at the 
earth's surface, we have by (4) 



Q = 3-24. 10" -1-37. 10"'. ...... (6). 



Hence if the radial force were equal over the whole spherical 
surface, its amount would be 1 '37.10"" of terrestrial surface- 
gravity ; and every body on or near that surface would experience 
an acceleration toward the centre equal to 

1'37 . 10~ u kilometres per second per second ...... (7), 

because g is approximately 1000 centimetres per second per 
second, or '01 kilometre per second per second. If the normal 
force is not uniform, bodies on or near the spherical surface will 
experience centreward acceleration, some at more than that rate, 
some less. At exactly that rate, the velocity acquired per year 
(thirty-one and a half million seconds) would be 4*32 . 10~ kilo- 
metres per second. With the same rate of acceleration through 
five million years the velocity would amount to 21*0 kilometres 
per second, if the body started from rest at our spherical surface ; 
and the space moved through in five million years would be 
17 . lO 16 kilometres, which is only '055 of r (5). This is so small 
that the force would vary very little, unless through the accident 
of near approach to some other body. With the same acceleration 
constant through twenty-five million years the velocity would 
amount to 108 kilometres per second; but the space moved through 
in twenty-five million years would be 4'25 . 10 1 " kilometres, or 
more than the radius r, which shows that the rate of acceleration 
could not be approximately constant for nearly as long a time as 
twenty-five million years. It would, in fact, have many chances 
of being much greater than 108 kilometres per second, and many 
chances also of being considerably less. 

15. Without attempting to solve the problem of finding 
the motions and velocities of the thousand million bodies, we can 
see that if they had been given at rest* twenty-five million years 

* " The potential energy of gravitation may be in reality the ultimate created 



00 LECTURE XVI. 

Jolar. i*g distributed uniformly or n on -uniformly through our sphere (5) 
of 3'09 . 10 1 * kilometres radius, a very large proportion of them 
would now have velocities not less than twenty or thirty kilometres 
per second, while many would have velocities less than that; and 
certainly some would have velocities greater than 108 kilometres 
per second: or if thousands of millions of years ago they had 
been given at rest, at distances from one another very great in 
comparison with r (5), so distributed that they should temporarily 
now be equably spaced throughout a spherical surface of radius / 
(5), their mean velocity (reckoned as the square root of the mean 
of the squares of their actual velocities) would now be 50'4 kilo- 
metres per secund*. This is not very unlike what we know of 
the stars visible to us. Thus it is quite possible, perhaps pro- 
bable, that there may be as much matter as a thousand million 
suns within the distance corresponding to parallax one one- 
thousandth of a second (3'09 . 10 l * kilometres). But it seems 
perfectly certain that there cannot be within this distance as 
much matter as ten thousand million suns : because if there were, 
we should find much greater velocities of visible stars than 
observation shows : according to the following tables of results, 
and statements, from the most recent scientific authorities on the 
subject. 

antecedent of all the motion, heat, and light at present in the universe." See 
Mechanical Antecedent* of Motion, Heat, ami Li'tht, Art. LXTX. of my Collected 
Math, and Phys. Papers, Vol. 11. 

* To prove this, remark that the exhaustion of gravitational energy 



Thomson and Tail's Natural Philowphy, Part II. 549) when a vast number. .V, of 
equal masses come from rest at infinite distances from one another to an equably 
spaced distribution through a sphere of radius r i< easily found to be 3,10 Fr, 
where F denotes the resultant force of the attraction of all of them on a material 
point, of mass equal to the sum of their masses, placed at the spherical surface. 
Now this exhaustion of gravitational energy is spent wholly in the generation of 

kinetic energy ; and therefore we have 2 1 mr- = ^ fV, and by (7) F=1'37 . 10-2iN; 

whence 

5^ = 1 1-37. 10-.r 
2,m o 

which, for the case of equal masses, gives, with (5) for the value of r, 

v/^J- = %'l* 1-37 10 ~' 3 - 3 ' 09 10 18 ) = 50-4 kilometres per second. 



VELOCITIES OF STARS. 



Molar. 



From the Annuaire du Bureau des Longitudes (Paris, 1901). 



M t ' u jf Name of Star 


Distance 
from earth, 
in million 
million 
Kilometres 


Annual 
proper 
motions 


Velocities 
perpendicular 

metres per 
second 


0*7 a Centanri 


43 


3-62 


0-72 23-9 


6-8 21185 Lalande 


64 


4-75 


0-48 47-1 


5'1 61 Cygni 


70 


5-17 


0-44 55-7 


-1*4 Sirius 


83 


1-32 


0-37 17-0 


8-2 18609 Arg.-OCltzen... 
7-9 34 Groombridge 
7'5 9352 Lacaille 
0'5 Procyon 
9-0 i 11677 Arg.-(Eltzen... 
5 1643 Fedorenko 
8-5 21258 Lalande 
4*7 ff Draconis 
3*6 i> Cassiopeia 
0-2 a Auriga 
90 17415 Arg.-(Eltzen... 
0*9 a Aquihe 


88 
99 
110 
110 
119 
123 
128 
128 
147 
147 
154 
154 


2-30 

2-as 

697 
1-26 
3-05 
1-43 
4-40 
1-84 
1-19 
0-43 
1-27 
0-64 


0-35 31-3 
0-31 43-5 
0-28 118-5 
0-27 22-2 
0-26 55-7 
025 27-2 
0-24 87-1 
024 36-5 
0-21 27-0 
021 9-8 
0-20 3fr2 
0-20 15-2 


5-2 e "Indien" 


154 


4-60 


0-20 109-5 


4-5 o 2 feridani 


181 


4*05 


0-17 113-2 


2*4 /3 Cassiopeia 


193 


057 


0-16 169 


1*0 a Tauri 


M 


0-19 


0-15 6-0 


7-0 1831 Fedorenko 
4-1 j/Ophiuchi 
0-2 Vega 
2-2 a Urs. Min. (Polaris) 


206 
206 
206 
440 


0-42 
1-13 
0-36 
0-05 


015 13-3 
0-15 35-8 
0-15 11-4 
0-07 3-4 



Stars which have largest of observed Velocities in the Line of 
Sight. (Extract by the Astronomer Royal from an Article 
in the Astrophysical Journal for 1901, January, by W. W. 
Campbell, Director of Lick Observatory.) 



?- 


8* 


B.A. 


Deo. 


Velocity 


4-6 


e Andromeda 
H Cassiopeiae 


h. m. 
33 
1 


+ 28 46 
+ 54 20 


- 84 km. per sec. 
- 97 




S Leporis 


5 47 


-20 54 


+ 95 i. 


4-2 


e Canis Majoris 
t Pegasi 


6 50 
21 17 


-11 55 
+ 19 23 


+ % 
- 76 


4-1 


M Sagittarii 


18 8 


-21 1 


- 76 













The + sign denotes recession, the - sign approach. 



272 



LECTURE XVI. 



Molar. Motions of Stars in the Line of Sight determined at Potsdam 
Observatory, 1889-1891. (Communicated by Professor Becker, 
University Observatory, Glasgow.) 



Star 


Magni- 
tude 


Velocity 
relative to 
the Sun 


Star 


Magni- 
tude 


Velocity 
relative to 
the Sun 






km. 






km. 


a Andromeda ... 


2-0 


+ 4-5 


7 Leonis 


2-0 


-38-5 


ft Cassiopeiae 


2-1 


-f 5-2 


/3 Ursa? Majoris . . 


2-3 


-29-3 


a Cassiopeia 


var. 


-15-2 


o Ursae Majoris . . 


2-0 


-11-9 


7 Cassiopeia; 


2-0 


- 3-5 


5 Leonis 


2-3 


-14-4 


/3 Andromeda? ... 


2-3 


+ 11-2 


j3 Leonis 


2-0 


-12-2 


a Ursae Minoris ... 


2-0 


-25-9 


7 Ursae Majoris . . 


2-3 


-26-6 


7 Andromeda? , . . 


2-4 


-12-9 


e Ursa? Majoris . . 


2-0 


-30-3 


a Arietis 


2-0 


-14-7 


a Virginis 


1-0 


-14-8 


/3 Persei 


var. 


- 1-5 


f UrsaB Majoris . . 


2-1 


-31-2 


a Persei 


2-0 


-10-3 


i\ Ursae Majoris . . 


2-0 


-26-2 


a Tauri 


1-0 


+ 48-5 


a Bootis 


1-0 


- 7-7 


a Auriga? 


1-0 


+ 24-5 


e Bootis 


2-0 


-16-3 


/3 Orionis 


1-0 


+ 16-4 


|8 Ursae Minoris . . 


2-0 


+ 14-2 


y Orionis 


2-0 


+ 9-2 


|8 Librae 


2-0 


- 9-6 


p Tauri 


2-0 


+ 8-0 


a Coronas 


2-0 


+ 32-0 


5 Orionis 


2-5 


+ 0-9 


a Serpentis 


2-3 


+ 22-3 


e Orionis 


2-0 


+ 26-5 


ft Herculis 


2-3 


-35-3 


f Orionis 


2-0 


+ 14-8 


a Opliiuchi 


2-0 


+ 19-2 


a Orionis 


var. 


+ 17-2 


a Lyrae 


1-0 


-15-3 


/j Auriga? 


2-0 


- 28-1 


a Aciuiltc 


1-3 


-36-9 


7 Geminorum ... 


2-3 


-16-6 


7 Cygni 


2-4 


- G-4 


a Canis Majoris... 


1-0 


-15-6 


a Cygni 


l-<5 


- 8-0 


a Geminorum ... 


2-3 


- 29-7 


e Pegasi 


2-3 


+ 8-0 


a Canis Minoris... 


1-0 


- 9-2 


/SPegasi 


var 


+ 6-7 


/3 Geminorum 


1-3 


+ 1-1 


a Pegasi 


2-0 


1 1-3 


a Leonis 


1-3 


- 9-1 





















The velocity of the sun relatively to stars in general according 
to Kempf and Risteen is probably about 19 kilometres per second*. 
In respect to greatest proper motions and velocities Sir Norman 
Lockyer gives me the following information: "The star with 
" the greatest known proper motion (across the line of sight) is 
" 243 Cordoba = 8"'7 per annum. Velocity in kilometres not 
" known. 

" 1830 Groombridge has a proper motion of 7"'0 per annum 
" and a parallax of 0"*089, from which it results that the velocity 
" across the line of sight is 370 kms. per second. Various esti- 
" mates of the parallax, however, have been made and this velocity 
" is somewhat uncertain. The star with the greatest known 
" velocity in the line of sight is Herculis, which travels at 
" 70 kms. per second. 

* See footnote on 10 of Appendix B. 



VELOCITIES OF STARS. 273 

" The dark Hue component of Nova Persei was approaching Molar, 
"the earth with a velocity of over 1100 kms. per second." This 
last-mentioned and greatest velocity is probably that of a torrent 
of gas due to comparatively small particles of melted and evaporat- 
ing fragments shot out laterally from two great solid or liquid 
masses colliding with one another, which may ^ lany times 
greater than the velocity of either before col Us 1 ' >t as we see 

in the trajectories of small fragments shot c^u nearly horizontally 
when a condemned mass of cast-iron is broken up by a heavy 
mass of iron falling upon it from a height of perhaps twenty feet 
in engineering works. 

16. Newcomb has given a most interesting speculation 
regarding the very great velocity of 1830 Groombridge, which he 
concludes as follows: "If, then, the star in question belongs to 
" our stellar system, the masses or extent of that system must 
"be many times greater than telescopic observation and astro- 
" nomical research indicate. We may place the dilemma in a 
" concise form, as follows : 

" Either the bodies which compose our universe are vastly 
" more massive and numerous than telescopic examination seems 
" to indicate, or 1830 Groombridge is a runaway star, flying on a 
" boundless course through infinite space with such momentum that 
" the attraction of all the bodies of the universe can never stop it. 

" Which of these is the more probable alternative we cannot 
" pretend to say. That the star can neither be stopped, nor bent 
" far from its course until it has passed the extreme limit to 
" which the telescope has ever penetrated, we may consider 
" reasonably certain. To do this will require two or three millions 
" of years. Whether it will then be acted on by attractive forces 
"of which science has no knowledge, and thus carried back to 
" where it started, or whether it will continue straightforward for 
" ever, it is impossible to say. 

" Much the same dilemma may be applied to the past history 
" of this body. If the velocity of two hundred miles or more per 
" second with which it is moving exceeds any that could be pro- 
" duced by the attraction of all the other bodies in the universe, 
" then it must have been flying forward through space from the 
" beginning, and, having come from an infinite distance, must be 
" now passing through our system for the first and only time." 
T.L. 18 



274 LECTURE XVI. 

Molar. | 17. In all these views the chance of passing another star at 

some small distance such as one or two or three times the sun's 
radius has been overlooked; and that this chance is not excessively 
rare seems proved by the multitude of Novas (collisions and their 
sequels) known in astronomical history. Suppose, for example, 
1830 Groombridge, moving at 370 kilometres per second, to chase 
a star of twenty times the sun's mass, moving nearly in the same 
direction with a velocity of 50 kilometres per second, and to 
overtake it and pass it as nearly as may be without collision. Its 
own direction would be nearly reversed and its velocity would be 
diminished by nearly 100 kilometres per second. By two or three 
such casualties the greater part of its kinetic energy might be 
given to much larger bodies previously moving with velocities of 
less than 100 kilometres per second. By supposing reversed, the 
motions of this ideal history, we see that 1830 Groombridge may 
have had a velocity of less than 100 kilometres per second at some 
remote past time, and may have had its present great velocity 
produced by several cases of near approach to other bodies of much 
larger mass than its own, previously moving in directions nearly 
opposite to its own, and with velocities of less than 100 kilometres 
per second. Still it seems to me quite possible that Newcomb's 
brilliant suggestion may be true, and that 1830 Groombridge is a 
roving star which has entered our galaxy, and is destined to travel 
through it in the course of perhaps two or three million years, and 
to pass away into space never to return to us. 

18. Many of our supposed thousand million stars, perhaps 
a great majority of them, may be dark bodies ; but let us suppose 
for a moment each of them to be bright, and of the same size and 
brightness as our sun ; and on this supposition and on the further 
suppositions that they are uniformly scattered through a sphere 
(5) of radius 3'09 . 10 18 kilometres, and that there are no stars out- 
side this sphere, let us find what the total amount of starlight 
would be in comparison with sunlight. Let n be the number per 
unit of volume, of an assemblage of globes of radius a scattered 
uniformly through a vast space. The number in a shell of radius 
q and thickness dq will be n . 4>7rq z dq, and the sum of their 
apparent areas as seen from the centre will be 

n . 4>Trq-dq or n . 4?7r z ardq. 



TOTAL APPARENT AREA OF STARS. 275 

Hence by integrating from q = to q = r we find Molar. 

n.4-7ra*r (8) 

for the sum of their apparent areas. Now if N be the total number 
in the sphere of radius r we have 



Hence (8) becomes N. STT f -J ; and if we denote by a the ratio of 
the sum of the apparent areas of all the globes to 4?r we have 



(1 - a)/, very approximately equal to I/a, is the ratio of the 
apparent area not occupied by stars to the sum of the apparent 
areas of all their discs. Hence a is the ratio of the apparent 
brightness of our star-lit sky to the brightness of our sun's disc. 
Cases of two stars eclipsing one another wholly or partially would, 
with our supposed values of r and a, be so extremely rare that 
they would cause a merely negligible deduction from the total of 
(10), even if calculated according to pure geometrical optics. This 
negligible deduction would be almost wholly annulled by diffraction, 
which makes the total light from two stars of which one is eclipsed 
by the other, very nearly the same as if the distant one were seen 
clear of the nearer. 

19. According to our supposition of 18, we have N= 10*. 
a = 7.10 s kilometres, and therefore r/o = 4'4 . 10'. Hence 
by (10) 

a = 3-87.10-" ........................... (11). 

This exceedingly small ratio will help us to test an old and 
celebrated hypothesis that if we could see far enough into space 
the whole sky would be seen occupied with discs of stars all of 
perhaps the same brightness as our own sun, and that the 
reason why the whole of the night-sky and day-sky is not as 
bright as the sun's disc is that light suffers absorption in 
travelling through space. Remark that if we vary r keeping 
the density of the matter the same, JV varies as the cube of r. 
Hence by (10) a varies simply as ?; and therefore to make a 
even as great as 387/100, or, say, the sum of the apparent 

182 



276 LECTURE XVI. 

areas of discs 4 per cent, of the whole sky, the radius must be 
10". r or 3'09. 10 27 kilometres. Now light travels at the rate 
of 300,000 kilometres per second or 9'45.10 12 kilometres per year. 
Hence it would take 3'27.10 14 or about 3J.10 14 years to travel 
from the outlying suns of our great sphere to the centre. Now 
we have irrefragable dynamics proving that the whole life of 
our sun as a luminary is a very moderate number of million 
years, probably less than 50 million, possibly between 50 and 100. 
To be very liberal, let us give each of our stars a life of a hundred 
million years as a luminary. Thus the time taken by light to 
travel from the outlying stars of our sphere to the centre would 
be about three and a quarter million times the life of a star. 
Hence, if all the stars through our vast sphere commenced 
shining at the same time, three and a quarter million times the 
life of a star would pass before the commencement of light 
reaching the earth from the outlying stars, and at no one 
instant would light be reaching the earth from more than an 
excessively small proportion of all the stars. To make the whole 
sky aglow with the light of all the stars at the same time the 
commencements of the different stars must be timed earlier 
and earlier for the more and more distant ones, so that the time 
of the arrival of the light of every one of them at the earth 
may fall within the durations of the lights at the earth of all 
the others ! Our supposition of uniform density of distribution 
is, of course, quite arbitrary; and ( 13, 15 above) we ought, in 
the greater sphere of 19, to assume the density much smaller 
than in the smaller sphere (5) ; and in fact it seems that there 
may not be enough of stars (bright or dark) to make a total of 
star-disc-area more than 10~ 12 or 10~ n of the whole sky. See 
Appendix D, " On the Clustering of Gravitational Matter in any 
" part of the Universe." 

20. To understand the sparseness of our ideal distribution 
of 1000 million suns, divide the total volume of the supposed 
sphere of radius r (5) by 10', and we find 123'S.IO 39 cubic kilo- 
metres as the volume per sun. Taking the cube root of this 
we find 4*98 . 10 13 kilometres as the edge of the corresponding 
cube. Hence if the stars were arranged exactly in cubic order 
with our sun at one of the eight corners belonging to eight 
neighbouring cubes, his six nearest neighbours would be each 



NUMBER OF VISIBLE STARS. 277 

at distance t'98.10" kilometres; which is the distance corre- Molar, 
spending to parallax 0" 62. Our sun seen at so great a distance 
would probably be seen as a star of something between the first 
and second magnitude. For a moment suppose each of our 
1000 million suns, while of the same mass as our own sun, to 
have just such brightness as to make it a star of the first magni- 
tude at distance corresponding to parallax 1"'0. The brightness 
at distance r (5) corresponding to parallax 0"'001 would be one 
one-millionth of this, and the most distant of our assumed 
stars would be visible through powerful telescopes as stars of the 
sixteenth magnitude. Newcomb (Popular Astronomy, 1883, 
p. 424) estimated between 30 and 50 million as the number of 
stars visible in modern telescopes. Young (General Astronomy, 
p. 448) goes beyond this reckoning and estimates at 100 million 
the total number of stars visible through the Lick telescope. 
This is only the tenth of our assumed number. It is never- 
theless probable that there may be as many as 1000 million 
stars within the distance r (5); but many of them may be 
extinct and dark, arid nine-tenths of them though not all dark 
may be not bright enough to be seen by us at their actual 
distances. 

21. I need scarcely repeat that our assumption of equable 
distribution is perfectly arbitrary. How far from being like the 
truth is illustrated by Herschel's view of the form of the universe 
as shown in Newcomb's Popular Astronomy, p. 469. It is quite 
certain that the real visible stars within the distance r (5) from 
us are very much more crowded in some parts of the whole 
sphere than in others. It is also certain that instead of being 
all equally luminous as we have taken them, they differ largely 
in this respect from one another. It is also certain that the 
masses of some are much greater than the masses of others ; 
as will be seen from the following table, which has been compiled 
for me by Professor Becker from Andrews Traiti d' Astronomic 
Stellaire, showing the sums of the masses of the components of 
some double stars, and the data from which these have been 
determined. 



278 



LECTURE XVI. 



1 










| Major axis 








Parallax 




in terms of 


Period, 
in 


in units 
of the 






in seconds 


semi-major 
axis of 


years 


sun's mass 








earth's orbit 






a Centauri 


: 75 


18-17 


25 


84 2-0 


61 Cygni 0-44 ! 29-48 


68 783 0-5 


Sirius 0-39 8'31 


24 52 


3-2 


Procyon 0'27 5-84 


4 40 6-3 


crEridani 0-19 5-72 


28 176 


0-9 


t) Cassiopeiaa 0-15 8-20 


39 : 190 


4-3 


p Ophiuchi 0-15 4-60 


30 88 


3-6 


7 Virginis 0-05 1 3-99 


79* 


194 


15-0 


7Leonis j 0-02 f 


1-98 


102* ! 407 


6-5 


1 


1 





22. There may also be a large amount of matter in many 
stars outside the sphere of 3.10 1(i kilometres radius, but however 
much matter there may be outside it, it seems to be made highly 
probable by 11 21, that the total quantity of matter within 
it is greater than 100 million times, and less than 2000 million 
times, the sun's mass. 

I wish, in conclusion, to express my thanks to Sir Norman 
Lockyer, to the Astronomer Royal Mr Christie, to Sir Robert 
Ball, and to Prof. Becker, for their kindness in taking much 
trouble to give me information in respect to astronomical data, 
which has proved most useful to me in 11 21 above. 

* From spectroscopic observations by Belopolsky of Poulcowa, combined with 
elements of orbit. 

t Parallax calculated from dynamical determinations of ratio of semi-major 
axis of double-star's orbit to semi-major axis of earth's orbit. 



LECTURE XVII. 

THURSDAY, October 16, 3.30 P.M. Altered (1901, 1902) to 
extension of Lee. XVI. 

23. HITHERTO in all our views we have seen nothing of abso- Molecular, 
lute dimensions in molecular structure, and have been satisfied to 
consider the distance between neighbouring molecules in gases, 
or liquids, or crystals, or non-crystalline solids to be very small in 
comparison with the shortest wave-length of light with which we 
have been concerned. Even in respect to dispersion, that is to 
say, difference of propagational velocity for different wave-lengths, 
it has not been necessary for us to accept Cauchy's doctrine that 
the spheres of molecular action are comparable with the wave- 
length. We have seen that dispersion can be, and probably in 
fact is, truly explained by the periods of our waves of light being 
not infinitely great in comparison with some of the periods of 
molecular vibration ; and, with this view, the dimensions of 
molecular structure might, so far as dispersion is concerned, be as 
small as we please to imagine them, in comparison with wave- 
lengths of light. Nevertheless it is exceedingly interesting and 
important for intelligent study of molecular structures and the 
dynamics of light, to have some well-founded understanding in 
respect to probable distances between centres of neighbouring 
molecules in all kinds of ponderable matter, while for the present 
at all events we regard ether as utterly continuous and structure- 
less. It may be found in some future time that ether too has a 
molecular structure, perhaps much finer than any structure of 
ponderable matter ; but at present we neither see nor imagine 
any reason for believing ether to be other than continuous and 
homogeneous through infinitely small contiguous portions of 
space void of other matter than ether. 

24. The first suggestion, so far as we now know, for estimat- 
ing the dimensions of molecular structure in ordinary matter was 



280 LECTURE XVII. 

Molecular, given in 1805 by Thomas Young*, as derived from his own and 
Laplace's substantially identical theories of capillary attraction. 
In this purely dynamical theory he found that the range of the 
attractive force of cohesion is equal to 3T/K] where T denotes 
the now well-known Young's tension of the free surface of a 
liquid, and K denotes a multiple integral which appears in 
Laplace's formulas and is commonly now referred to as Laplace's 
K, as to the meaning of which there has been much controversy 
in the columns of Nature and elsewhere. Lord Rayleigh in his 
article of 1890, " On the Theory of Surface Forcesf ," gives the 
following very interesting statement in respect to Young's estimate 
of molecular dimensions : 

25. " One of the most remarkable features of Young's treatise 
" is his estimate of the range a of the attractive force on the basis 
" of the relation T = ^aK. Never once have I seen it alluded to ; 
" and it is, I believe, generally supposed that the first attempt of 
" the kind is not more than twenty years old. Estimating K at 
" 23000 atmospheres, and T at 3 grains per inch, Young finds that 
" ' the extent of the cohesive force must be limited to about the 
"'250 millionth of an inch [10~ 8 cm.]'; and he continues, 'nor is 
" ' it very probable that any error in the suppositions adopted can 
" ' possibly have so far invalidated this result as to have made it 
"'very many times greater or less than the truth' Young con- 
" tinues : ' Within similar limits of uncertainty, we may obtain 
" ' something like a conjectural estimate of the mutual distance 
" ' of the particles of vapours, and even of the actual magnitude 
" ' of the elementary atoms of liquids, as supposed to be nearly in 
" ' contact with each other ; for if the distance at which the force 
" ' of cohesion begins is constant at the same temperature, and if 
" ' the particles of steam are condensed when they approach within 
" ' this distance, it follows that at 60 of Fahrenheit the distance 
" ' of the particles of pure aqueous vapour is about the 250 
" ' millionth of an inch ; and since the density of this vapour is 
" ' about one sixty thousandth of that of water, the distance of the 
" ' particles must be about forty times as great ; consequently the 
" ' mutual distance of the particles of water must be about the 

* " On the Cohesion of Fluids," Phil. Traits. 1805; Collected Works, Vol. i. p. 461. 
t Phil. Mag. Vol. xxx. 1890, p. 474. 



EARLIEST ESTIMATES OF MOLECULAR DIMENSIONS. 281 

"'ten thousand millionth of an inch* ['025 x 10~ 8 cm.]. It is Molecular, 

" ' true that the result of this calculation will differ considerably 

"'according to the temperature of the substances compared.... 

" ' This discordance does not however wholly invalidate the general 

" 'tenour of the conclusion... and on the whole it appears tolerably 

" ' safe to conclude that, whatever errors may have affected the 

" ' determination, the diameter or distance of the particles of 

" ' water is between the two thousand and the ten thousand 

" ' millionth of an inch' [between 125 x 10~* and '025 x 10~ 8 of a 

" cm.]. This passage, in spite of its great interest, has been so 

" completely overlooked that I have ventured briefly to quote it, 

" although the question of the size of atoms lies outside the scope 

" of the present paper." 

26. The next suggestion, so far as I know, for estimating the 
dimensions of molecular structure in ordinary matter, is to be 
found in an extract from a letter of my own to Joule on the 
contact electricity of metals, published in the Proceedings of the 
Manchester Literary aud Philosophical Society f, Jan. 21, 1862, 
which contains the following passage : " Zinc and copper con- 
" nected by a metallic arc attract one another from any distance. 
" So do platinum plates coated with oxygen and hydrogen respec- 
" tively. I can now tell the amount of the force, and calculate 
" how great a proportion of chemical affinity is used up electrically, 
" before two such discs come within 1/1000 of an inch of one 
" another, or any less distance down to a limit within which 
" molecular heterogeneousness becomes sensible. This of course 
" will give a definite limit for the sizes of atoms, or rather, as I do 
" not believe in atoms, for the dimensions of molecular structures." t 
The theory thus presented is somewhat more fully developed in a 
communication to Nature in March 1870, on "The Size of Atoms +," 
and in a Friday evening lecture to the Royal Institution on the 

* Tonng here, curiously insensible to the kinetic theory of gases, supposes the 
molecules of vapour of water at 60 Fahr. to be within touch (or direct mutual 
action) of one another; and thus arrives at a much finer-grainedness for liquid 
water than he would have found if he had given long enough free paths to molecules 
of the vapour to account for its approximate fulfilment of Boyle's law. 

t Reproduced as Art. 22 of my Electrostatics and Magnetism. 

Rpublished as Appendix (F) in Thomson and Tait's Natural Philosophy, 
Part ii. Second Edition. 

Republished in Popular Lecture* and Addresses, Vol. i. 



282 LECTURE XVII. 

lecular. same subject on February 3, 1883; but to illustrate it, information 
was wanting regarding the heat of combination of copper and 
zinc. Experiments by Professor Roberts Austen and by Dr A. 
Gait, made within the last four years, have supplied this want ; 
and in a postscript of February 1898 to a Friday evening lecture 
on " Contact Electricity," which I gave at the Royal Institution 
on May 21, 1897, I was able to say "We cannot avoid seeing 
"molecular structures beginning to be perceptible at distances of 
" the hundred-millionth of a centimetre, and we may consider it 
" as highly probable that the distance from any point in a molecule 
" of copper or zinc to the nearest corresponding point of a neighbour- 
" ing molecule is less than one one-hundred-millionth, and greater 
"than one one-thousand-millionth of a centimetre"; and also to 
confirm amply the following definite statement which I had 
given in my Nature article (1870) already referred to: "Plates 
" of zinc and copper of a three hundred-millionth of a centimetre 
" thick, placed close together alternately, form a near approxima- 
" tion to a chemical combination, if indeed such thin plates could 
" be made without splitting atoms." 

27. In that same article thermodynamic considerations in 
stretching a fluid film against surface tension led to the following 
result : " The conclusion is unavoidable, that a water-film falls 
" off greatly in its contractile force before it is reduced to a thick- 
"ness of a two hundred-millionth of a centimetre. It is scarcely 
" possible, upon any conceivable molecular theory, that there can 
" be any considerable falling off in the contractile force as long as 
" there are several molecules in the thickness. It is therefore 
" probable that there are not several molecules in a thickness of a 
" two-hundred-millionth of a centimetre of water." More detailed 
consideration of the work done in stretching a water-film led me 
in my Royal Institution Lecture of 1883 to substitute one one- 
hundred-millionth of a centimetre for one two-hundred-millionth 
in this statement. On the other hand a consideration of the 
large black spots which we now all know in a soap-bubble or soap- 
film before it bursts, and which were described in a most interest- 
ing manner by Newton*, gave absolute demonstration that the 
film retains its tensile strength in the black spot "where the 

* Newton's Optics, pp. 187, 191, Edition 1721, Second Book, Part i.: quoted in 
my Royal Institution Lecture, Pop. Lectures and Addresses, Vol. i. p. 175. 



REINOLD AND RUCKER, RONTGEN, RAYLEIGH. 283 

" thickness is clearly much less than 1/60000 of a centimetre, Moleculai 
" this being the thickness of the dusky white" with which the 
black spot is bordered. And further in 1883 Reinold and 
Riicker's* admirable application of optical and electrical methods 
of measurement proved that the thickness of the black film in 
Plateau's " liquide glyce'rique" and in ordinary soap solution is 
between one eight-hundred-thousandth of a centimetre and one 
millionth of a centimetre. Thus it was certain that the soap-film 
has full tensile strength at a thickness of about a millionth of a 
centimetre, and that between one millionth and one one-hundred- 
millionth the tensile strength falls off enormously. 

28. Extremely interesting in connection with this is the 
investigation, carried on independently by Rontgen ) and Ray- 
leigh* and published by each in 1890, of the quantity of oil 
spreading over water per unit area required to produce a sensible 
disturbance of its capillary tension. Both experimenters ex- 
pressed results in terms of thickness of the film, calculated as if 
oil were infinitely homogeneous and therefore structureless, but 
with very distinct reference to the certainty that their films were 
molecular structures not approximately homogeneous. Rayleigh 
found that olive oil, spreading out rapidly all round on a previously 
cleaned surface of water from a little store carried by a short 
length of platinum wire, produced a perceptible effect on little 
floating fragments of camphor at places where the thickness of 
the oil was 10'6 x 10~ 8 cm., and no perceptible effect where the 
thickness was 8'1 x 10~ 8 cm. It will be highly interesting to 
find, if possible, other tests (optical or dynamical or electrical or 
chemical) for the presence of a film of oil over water, or of films 
of various liquids over solids such as glass or metals, demonstrat- 
ing by definite effects smallei and smaller thicknesses. Rontgen, 
using ether instead of camphor, found analogous evidence of layers 
5'6 x 10~* cm. thick. It will be very interesting for example to 
make a thorough investigation of the electric conductance of a 
clean rod of white glass of highest insulating quality surrounded 
by an atmosphere containing measured quantities of vapour of 

* " On the Limiting Thickness of Liquid Films," Roy. Soc. Proc. April 19, 1883; 
Phil. Trans. 1883, Part 11. p. 645. 

t Wied. Ann. Vol. xu. 1890, p. 321. 

J Proc. Roy. Soc. Vol. XLVII. 1890, p. 364. 



284 LECTURE XVII. 

decular. water. When the glass is at any temperature above the dew- 
point of the vapour, it presents, so far as we know, no optical 
appearance to demonstrate the pressure of condensed vapour of 
water upon it : but enormous differences of electric conductance, 
according to the density of the vapour surrounding it, prove the 
presence of water upon the surface of the glass, or among the 
interstices between its molecules, of which electric conductance 
is the only evidence. Rayleigh has himself expressed this view in a 
recent article, " Investigations on Capillarity" in the Philosophical 
Magazine* From the estimates of the sizes of molecules of argon, 
hydrogen, oxygen, carbonic oxide, carbonic acid, ethylene (C 2 H 4 ), 
and other gases, which we shall have to consider ( 47 below), we 
may judge that in all probability if we had eyes microscopic 
enough to see atoms and molecules, we should see in those thin 
films of Rayleigh and Rontgen merely molecules of oil lying 
at greater and less distances from one another, but at no part 
of the film one molecule of oil lying above another or resting 
on others. 

29. A very important and interesting method of estimating 
the size of atoms, founded on the kinetic theory of gases, was 
first, so far as I know, thought of by Loschmidt'f' in Austria and 
Johnstone Stoney in Ireland. Substantially the same method 
occurred to myself later and was described in Nature, March 1870, 
in an article^ on the " Size of Atoms " already referred to, 26 
above, from which the quotations in 29, 30 are taken. 

" The kinetic theory of gases suggested a hundred years ago 
" by Daniel Bernoulli has, during the last quarter of a century, 
" been worked out by Herapath, Joule, Clausius, and Maxwell 
" to so great perfection that we now find in it satisfactory ex- 
planations of all non-chemical" and non-electrical "properties of 
" gases. However difficult it may be to even imagine what kind 
" of thing the molecule is, we may regard it as an established 
" truth of science that a gas consists of moving molecules dis- 
" turbed from rectilinear paths and constant velocities by collisions 
" or mutual influences, so rare that the mean length of nearly 

* Phil. Mag. Oct. 1899, p. 337. 

+ Sitzungsberichte of the Vienna Academy, Oct. 12, 1865, p. 395. 
J Reprinted aa Appendix (F) in Thomson and Tait's Natural Philosophy, 
Part ii. p. 499. 



LOSCHMIDT, JOHNSTONE STONEY, KELVIN. 285 

" rectilinear portions of the path of each molecule is many times Molecnli 
"greater than the average distance from the centre of each 
" molecule to the centre of the molecule nearest it at any time. 
" If, for a moment, we suppose the molecules to be hard elastic 
"globes all of one size, influencing one another only through 
" actual contact, we have for each molecule simply a zigzag path 
" composed of rectilinear portions, with abrupt changes of direc- 
tion But we cannot believe that the individual molecules 

"of gases in general, or even of any one gas, are hard elastic 
" globes. Any two of the moving particles or molecules must act 
" upon one another somehow, so that when they pass very near 
"one another they shall produce considerable deflexion of the 
" path and change in the velocity of each. This mutual action 
" (called force) is different at different distances, and must vary, 
"according to variations of the distance, so as to fulfil some 
" definite law. If the particles were hard elastic globes acting 
' upon one another only by contact, the law of force would be 
"...zero force when the distance from centre to centre exceeds 
" the sum of the radii, and infinite repulsion for any distance less 
" than the sum of the radii. This hypothesis, with its ' hard and 
'" fast' demarcation between no force and infinite force, seems to 
" require mitigation." Boscovich's theory supplies clearly the 
needed mitigation. 

30. To fix the ideas we shall still suppose the force absolutely 
zero when the distance between centres exceeds a definite limit, X ; 
but when the distance is less than X, we shall suppose the force 
to begin either attractive or repulsive, and to come gradually to 
a repulsion of very great magnitude, with diminution of distance 
towards zero. Particles thus defined I call Boscovich atoms. We 
thus call ^X the radius of the atom, and X its diameter. We 
shall say that two atoms are in collision when the distance 
between their centres is less than X. Thus "two molecules in 
"collision will exercise a mutual repulsion in virtue of which the 
" distance between their centres, after being diminished to a mini- 
" mum, will begin to increase as the molecules leave one another. 
" This minimum distance would be equal to the sum of the radii, 
" if the molecules were infinitely hard elastic spheres ; but in 
"reality we must suppose it to be very different in different 
" collisions." 



286 LECTURE XVII. 

ecular. 31. The essential quality of a gas is that the straight line 
of uniform motion of each molecule between collisions, called 
the free path, is long in comparison with distances between centres 
during collision. In an ideal perfect gas the free path would 
be infinitely long in comparison with distances between centres 
during collision, but infinitely short in comparison with any length 
directly perceptible to our senses ; a condition which requires the 
number of molecules in any perceptible volume to be exceedingly- 
great. We shall see tliat in gases which at ordinary pressures 
and temperatures approximate most closely, in respect to com- 
pressibility, expansion by heat, and specific heats, to the ideal 
perfect gas, as, for example, hydrogen, oxygen, nitrogen, carbon- 
monoxide, the free path is probably not more than about one 
hundred times the distance between centres during collisions, 
and is little short of 10~ 5 cm. in absolute magnitude. Although 
these moderate proportions suffice for the well-known exceedingly 
close agreement with the ideal gaseous laws presented by those 
real gases, we shall see that large deviations from the gaseous 
laws are presented with condensations sufficient to reduce the free 
paths to two or three times the diameter of the molecule, or to 
annul the free paths altogether. 

3:2. It is by experimental determinations of diffusivity that 
the kinetic theory of gases affords its best means for estimating 
the sizes of atoms or molecules and the number of molecules 
in a cubic centimetre of gas at any stated density. Let us 
therefore now consider carefully the kinetic theory of these 
actions, and with them also, the properties of thermal conductivity 
and viscosity closely related to them, as first discovered and 
splendidly developed by Clausius and Clerk Maxwell. 

33. According to their beautiful theory, we have three 
kinds of diffusion ; diffusion of molecules, diffusion of energy, and 
diffusion of momentum. Even in solids, such as gold and lead, 
Roberts-Austen has discovered molecular diffusion of gold into 
lead and lead into gold between two pieces of the metals when 
pressed together. But the rate of diffusion shown by this ad- 
mirable discovery is so excessively slow that for most purposes, 
.scientific and practical, we may disregard wandering of any 
molecule in any ordinary solid to places beyond direct influence of 



CLAUSIUS, MAXWELL. 287 

its immediate neighbours. In an elastic solid we have diffusion Moleculai 
of momentum by wave motion, and diffusion of energy consti- 
tuting the conduction of heat through it. These diffusions are 
effected solely by the communication of energy from molecule to 
molecule and are practically not helped at all by the diffusion of 
molecules. In liquids also, although there is thorough molecular 
diffusivity, it is excessively slow in comparison with the two other 
diffusivities, so slow that the conduction of heat and the diffusion 
of momentum according to viscosity are not practically helped by 
molecular diffusion. Thus, for example, the thermal diffusivity* 
of water ('002, according to J. T. Bottomley's first investigation, 
or about "001 of according to later experimenters) is several 
hundred times, and the diffusivity for momentum is from one to 
two thousand times, the ditfusivity of water for common salt, and 
other salts such as sulphates, chlorides, bromides, and iodides. 

34. We may regard the two motional diffusivities of a liquid 
as being each almost entirely due to communication of motion from 
one molecule to another. This is because every molecule is always 
under the influence of its neighbours and has no free path. When 
a liquid is rarefied, either gradually as in Andrew's experiments 
showing the continuity of the liquid and gaseous states, or 
suddenly as in evaporation, the molecules become less crowded 
and each molecule gains more and more of freedom. When the 
density is so small that the straight free paths are great in com- 
parison with the diameters of molecules, the two motional diffu- 
sivities are certainly due, one of them to carriage of energy, and 
the other to carriage of momentum, chiefly by the free rectilinear 
motion of the molecules between collisions. Interchange of 
energy or of momentum between two molecules during collision 
will undoubtedly to some degree modify the results of mere 
transport ; and we might expect on this account the motional 
diffusivities to be approximately equal to, but each somewhat 
greater than, the molecular diffusivity. If this view were correct, 
it would follow that, in a homogeneous gas when the free paths 
are long in comparison with the diameters of molecules, the 
viscosity is equal to the molecular diffusivity multiplied by the 

* Math, and Phyt. Papers, Vol. in. p. 226. For explanation regarding diffusi- 
vity and viscosity see same volume, pp. 428435. 

t See a paper by Milner and Chattock, Phil. Mag. Vol. ZLVIII. 1899. 



288 LECTURE XVII. 

lolecular. density, and the thermal conductivity is equal to the molecular 
diffusivity multiplied by the thermal capacity per unit bulk, 
pressure constant : and that whatever deviation from exactness of 
these equalities there may be, would be in the direction of the 
motional diffusivities being somewhat greater than the molecular 
diffusivity. But alas, we shall see, 45 below, that hitherto 
experiment does not confirm these conclusions : on the contrary 
the laminar diffusivities (or diffusivities of momentum) of the 
only four gases of which molecular diffusivities have been de- 
termined by experiment, instead of being greater than, or at 
least equal to, the density multiplied by the molecular diffusivity, 
are each somewhat less than three-fourths of the amount thus 
calculated. 

.35. I see no explanation of this deviation from what 
seems thoroughly correct theory. Accurate experimental deter- 
minations of viscosities, whether of gases or liquids, are easy by 
Graham's transpirational method. On the other hand even roughly 
approximate experimental determinations of thermal diffusivities 
are exceedingly difficult, and I believe none, on correct experi- 
mental principles, have really been made*; certainly none un- 
vitiated by currents of the gas experimented upon, or accurate 
enough to give any good test of the theoretical relation between 
thermal and material diffusivities, expressed by the following 
equation, derived from the preceding verbal statement regarding 
the three diffusivities of a gas, 

e=Kp=Kn = kcn, 

where 6 denotes the thermal conductivity, /t the viscosity, p the 
density, Kp the thermal capacity per unit bulk pressure constant, 
K the thermal capacity per unit mass pressure constant, c the 
thermal capacity per unit mass volume constant, and k the ratio 
of the thermal capacity pressure constant to the thermal capacity 
volume constant. It is interesting to remark how nearly theo- 

* So far as I know, all attempts hitherto made to determine the thermal con- 
ductivities of gases have been founded on observations of rate of communication of 
heat between a thermometer bulb, or a stretched metallic wire constituting an 
electric resistance thermometer, and the walls of the vessel enclosing it and the gas 
experimented upon. See WiedemannWwnakM, 1888, Vol. xxxiv. p. 023, and 1891, 
Vol. XLIV. p. 177. For other references, see 0. E. Meyer, 107. 



MOLECULAR DIFFUSIVITY. 289 

retical investigators* have come to the relation 0= A;c/*; Clausius Molecular. 
gave = %cp; O. E. Meyer, = l'6027c/i, and Maxwell, 0=fc/*. 
Maxwell's in fact is 6 = kcp for the case of a monatomic gas. 

36. To understand exactly what is meant by molecular 
diffusivity, consider a homogeneous gas between two infinite 
parallel planes, OGG and RRR, distance a apart, and let it be 
initially given in equilibrium ; that is to say, with equal numbers 
of molecules and equal total kinetic energies in equal volumes, 
and with integral of component momentum in any and every 
direction, null. Let N be the number of molecules per unit 
volume. Let every one of the molecules be marked either green 
or red, and whenever a red molecule strikes the plane GGO, let its 
marking be altered to green, and, whenever a green molecule 
strikes RRR, let its marking be altered to red. These markings 
are not to alter in the slightest degree the mass or shape or elastic 
quality of the molecules, and they do not disturb the equilibrium 
of the gas or alter the motion of any one of its particles; they 
are merely to give us a means of tracing ideally the history of any 
one molecule or set of molecules, moving about and colliding with 
other molecules according to the kinetic nature of a gas. 

37. Whatever may have been the initial distribution of the 
greens and reds, it is clear that ultimately there must be a regular 
transition from all greens at the plane GGG and all reds at the 
plane RRR, according to the law 



where g and r denote respectively the number of green molecules 
and of red molecules per unit volume at distance x from the 
plane RRR. In this condition of statistical equilibrium, the 
total number of molecules crossing any intermediate parallel 
plane from the direction GGG towards RRR will be equal to the 
number crossing from RRR towards GGG in the same time ; but 
a larger number of green molecules will cross towards RRR than 
towards GGG, and, by an equal difference, a larger number of red 
molecules will cross towards GGG than towards RRR. If we 
denote this difference per unit area per unit time by QN, we have 

* See the last ten lines of O. E. Meyer's book. 
T. L. 19 



290 LECTURE XVII. 

Molecular, for what I call the material diffusivity (called by Maxwell, "co- 
efficient of diffusion"), 

D = Qa (2). 

We may regard this equation as the definition of diffusivity. 
Remark that Q is of dimensions LT~\ because it is a number per 
unit of area per unit of time (which is of dimensions L~ 2 T~ l ) 
divided by N, a number per unit of bulk (dimensions L~ 3 ). Hence 
the dimensions of a diffusivity are L-T~ l ; and practically we 
reckon it in square centimetres per second. 

38. Hitherto we have supposed the G and the R particles 
to be of exactly the same quality in every respect, and the dif- 
fusivity which we have denoted by D is the inter-diffusivity of 
the molecules of a homogeneous gas. But we may suppose G arid 
R to be molecules of different qualities ; and assemblages of G 
molecules and of R molecules to be two different gases. Every- 
thing described above will apply to the inter-diffusions of these 
two gases ; except that the two differences which are equal when 
the red and green molecules are of the same quality are now not 
equal or, at all events, must not without proof be assumed to 
be equal. Let us therefore denote by Q ff N the excess of the 
number of G molecules crossing any intermediate plane towards 
RRR over the number crossing towards GGG, and by Q r N the 
excess of the number of R molecules crossing towards GGG above 
that crossing towards RRR. We have now two different diffusi vities 
of which the mean values through the whole range between the 
bounding planes are given by the equations 
D^Q^a; D r =Q r a- 

one of them, D g , the diffusivity of the green molecules, and the 
other, D,., the diffusivity of the red molecules through the hetero- 
geneous mixture in the circumstances explained in 37. We 
must not now assume the gradients of density of the two gases 
to be uniform as expressed by (1) of 37, because the homogeneous- 
ness on which these equations depend no longer exists. 

39. To explain all this practically*, let, in the diagram, the 
planes GGG, and RRR, be exceedingly thin plates of dry porous 
material such as the fine unglazed earthenware of Graham's experi- 

* For a practical experiment it might be necessary to allow for the difference of 
the proportions of the G gas on the two sides of the RRR plate and of the R gas 
on the two sides of the GGG plate. This would be exceedingly difficult, though 
not impossible, in practice. The difficulty is analogous to that of allowing for the 



INTER-DIFFUSION OF EQUAL AND SIMILAR MOLECULES. 291 



ments Instead of our green and red marked molecules of the same Molecular, 
kind, let us have two gases, which we shall call and R, supplied in 
abundance at the middles of the two ends of a non-porous tube of 



. 



electric resistances of the connections at the ends of a stout bar of metal of which 
it is desired to measure the electric resistance. But the simple and accurate 
"potential method" by which the difficulty is easily and thoroughly overcome 
in the electric case is not available here. I do not, however, put forward the 
arrangement described in the text as an eligible plan for measuring the inter- 
diffusivity of two gases. Even if there were no other difficulty, the quantities of the 
two pure gases required to realize it would be impracticably great. 

192 



292 LECTURE XVIT. 

Molecular, glass or metal, and guided to flow away radially in contact with 
the end-plates as indicated in the diagram. If the two axial 
supply-streams of the two pure gases are sufficiently abundant, 
the spaces GGG, RRR, close to the inner sides of the porous 
end-plates will be occupied by the gases G and R, somewhat nearly 
pure. They could not be rigorously pure even if the velocities of 
the scouring gases on the outer sides of the porous end-plates 
were comparable with the molecular velocities in the gases, and 
if the porous plates were so thin as to have only two or three 
molecules of solid matter in their thickness. The gases in contact 
with the near faces of the porous plates would, however, probably 
be somewhat approximately pure in practice with a practically 
realizable thinness of the porous plates, if a, the distance between 
the two plates, is not less than five or six centimetres and the 
scouring velocities moderately, but not impracticably, great. 
According to the notation of 37, Q g is the quantity of the G gas 
entering across GGG and leaving across RRR per sec. of time per 
sq. cm. of area ; Q r is the quantity of the R gas entering across 
RRR and leaving across GGG per sec. of time per sq. cm. of area ; 
the unit quantity of either gas being that which occupies a cubic 
centimetre in its entry tube. The equations 



where g and ? are the proportions of the G gas at R and of the R 
gas at G, define the average diffusivities of the two gases in the 
circumstances in which they exist in the different parts of the 
length a between the end-plates. This statement is cautiously 
worded to avoid assuming either equal values of the diffusivities 
of the two gases or equality of the diffusivity of either gas through- 
out the space between the end-plates. So far as I know difference 
of diffusivity of the two gases has not been hitherto suggested by 
any writer on the subject. What is really given by Loschmidt's 
experiments, 43 below, is the arithmetic mean of the two 
diffusivities D g and D r . 

40. In 1877 O. E. Meyer expressed the opinion on theoreti- 
cal grounds, which seem to me perfectly valid, that the inter- 
diffusivity of two gases varies according to the proportions of the 
two gases in the mixture. In the 1899 edition of his Kinetic 
Theory of Gases* he recalls attention to this view and quotes 
results of various experimenters, Loschmidt, Obermayer, Waitz, 
* Baynes' translation, p. 264. 



INTER-DIFFUSIVITY OF TWO GASES. 293 

seeming to support it, but, as he says, not quite conclusively. On Molecular. 
the other hand, Maxwell's theory ( 41 below) gives inter-diffu- 
sivity as independent of the proportions of the two gases; and 
only a single expression for diffusivity, which seems to imply that 
the two diffusivities are equal according to his theory. The 
subject is of extreme difficulty and of extreme interest, theoretical 
and practical ; and thorough experimental investigation is greatly 
to be desired. 

41. In 1873 Maxwell* gave, as a result of a theoretical 
investigation, the following formula which expresses the inter- 
diffusivity ( /),) of two gases independently of the proportion of the 
two gases in any part of the mixture: each gas being supposed to 
consist of spherical Boscovich atoms mutually acting according to 
the law, force zero for all distances exceeding the sum of the radii 
(denoted by , 2 ) and infinite repulsion when the distance between 
their centres is infinitely little less than this distance : 



where w lt w, are the masses of the molecules in the two gases 
in terms of that of hydrogen called unity ; V is the square root of 
the mean of the squares of the velocities of the molecules in 
hydrogen at C.; and N is the number of molecules in a cubic 
centimetre of a gas (the same for all gases according to Avogadro's 
law) at 0C. and standard atmospheric pressure. I find the 
following simpler formula more convenient 



where V*, V.? are the mean squares of the molecular velocities of 
the two gases at C., being the values of 'Apjp for the two gases, 
or three times the squares of their Newtonian velocities of sound, 
at that temperature. For brevity, we shall call mean molecular 
velocity the square root of the mean of the squares of the 
velocities of the molecules. The same formula is, of course, 
applicable to the molecular diffusivity of a single gas by taking 
F, = F 3 = V its mean molecular velocity, and s l3 = s the diameter 
of its molecules ; so that we have 



* " On Loschmidt's Experiments on Diffusion in relation to the Kinetic Theory 
of Gases," Mature, Aug. 1873 ; Scientific Papers, Vol. n. pp. 343 350. 



294 



LECTURE XVII. 



Molecular. 42. It is impossible by any direct experiment to find the 
molecular diffusivity of a single gas, as we have no means of 
marking its particles in the manner explained in 36 above; but 
Maxwell's theory gives us, in a most interesting manner, the 
means of calculating the diffusivity of each of three separate 
gases from three experiments determining the inter-diffusivities 
of their pairs. From the intcr-diffusivity of each pair determined 
by experiment we find, by (2) 41, a value of s 12 <y/(2 \/'37rN) for 
each pair, and we have s 12 = ^ (sj + s. 2 )* whence 

s 1 = 5 12 + 6- 13 - ^ ; s a = s ls + s. a - s a ; s 3 = s l3 + s. a - s 12 (1). 

Calculating thus the three values of s V(2 VS-TriV), and using 
them in (3) 41, we find the molecular diffusivities of the three 
separate gases. 

43. In two communications^ to the Academy of Science 
of Vienna in 1870, Loschmidt describes experimental determin- 
ations of the inter-diffusivities of ten pairs of gases made, by 
a well-devised method, with great care to secure accuracy. In 
each case the inter-diffusivity determined by the experiment 
would be, at all events, somewhat approximately the mean of the 
two diffusivities, 39 above, if these are unequal. The results 
reduced to C. and standard atmospheric pressure, and multi- 
plied by 2*78 to reduce from Loschmidt's square metres per hour 
to the now usual square centimetres per second, are as follows : 
TABLE OF INTER-DIFFUSIVITIES D. 



Pairs 


of gases 


D 
in sq. cms. per sec. 


Ho, 


2 


7214 


H,, 


CO 


C422 


H 2) 


CO, 


5558 


2 , 


CO 


1802 


2 , 


CO, 


1409 


CO, 


CO, 


1406 


co a 


, Air 


1423 


CO, 


, NO 


0984 


CO, 


, CH 4 


1587 


so, 


, H 3 


4809 



* This agrees with Maxwell's equation (4), but shows his equation (C) to be 
incorrect. 

t " Experimental-Untersuchungen uber die Diffusion von Gasen ohne porose 
Scheidfcwaude," Sitz. d. k. Akad. d. Wisciiach., March 10 and May 12, 1870. 



LOSCHMIDT'S DETERMINATIONS OF INTER-DIFFUSIVITIES. 295 



In the first six of these, each of the four gases H 2 , O 8 , CO, CO 2 Molecular, 
occurs three times, and we have four sets of three inter-diffusivities 
giving in all three determinations of the diffusivity of each gas as 
follows : 

Pairs of gases 

(12, 13, 23) 1-32 

(12, 14, 24) 1-35 

.1-26 
Gases 

H, (1) 

O, (2) 

CO (3) 

CO, (4) (12, 13, 23) -169 

(13, H, 34) 175 

(23, 24, 34) -178 



(13, 14, 34).. 

Mean 1-31 



Pairs of gases /) 

(12, 13, 23) -193 

(12, 14, 24) 190 

(23, 24, 34) -183 

Mean -189 



(12, 14, 24) -106 

(13, 14, 34) -Ill 

(23, 24, 34) -109 



Mean '174 



Mean "109 



Considering the great difficulty of the experimental investiga- 
tion, we may regard the agreements of the three results for each 
separate gas as, on the whole, very satisfactory, both in respect to 
the accuracy of Loschmidt's experiments and the correctness of 
Maxwell's theory. It certainly is a very remarkable achievement 
of theory and experiment to have found in the four means of the 
sets of three determinations, what must certainly be somewhat 
close approximations to the absolute values for the four gases, 
hydrogen, oxygen, carbon-monoxide, and carbon-dioxide, of some- 
thing seemingly so much outside the range of experimental 
observation, as the inter-diffusivity of the molecules of a separate 
gas. 

44. Maxwell, in his theoretical writings of different dates, 
gave two very distinct views of the inner dynamics of viscosity in 
a single gas, both interesting, and each, no doubt, valid. In one*, 
viscous action is shown as a subsidence from an " instantaneous 
rigidity of a gas." In the other^f*, viscosity is shown as a diffu- 
sion of momentum : and in p. 347 of his article quoted in 41 

* Tram. Roy. Soc., May 1866 ; Scientific Papen, Vol. n. p. 70. 

t "Molecules," a lecture delivered before the Brit. Assoc. at Bradford, Scientific 
Papers, Vol. n. p. 378. See also 0. E. Meyer's Kinetic Theory of Gate* (Baylies' 
trans. 189'J), 7476. 



296 



LECTURE XVII. 



Molecular, above he gives as from "the theory," but without demonstra- 
tion, a formula (5), which, taken in conjunction with (1), makes 



.(i); 



p denoting the density, p the viscosity, and D the molecular 
diffusivity, of any single gas. On the other hand, in his 1866 
paper he had given formulas making* 



.(2). 



45. Viewing viscosity as explained by diffusion of momentum 
we may, it has always seemed to me ( 34 above), regard (1) as 
approximately true for any gas, monatomic, diatomic, or polyatomic, 
provided only that the mean free path is large in comparison with 
the sum of the durations of the collisions. Unfortunately for this 
view, however, comparisons of Loschmidt's excellent experimental 
determinations of diffusivity with undoubtedly accurate determin- 
ations of viscosity from Graham's original experiments on trans- 
piration, and more recent experiments of Obermayer and other 
accurate observers, show large deviations from (1) and are much 
more nearly in agreement with (2). Thus taking -0000900, 
001430, '001234, "001974 as the standard densities of the four 
gases, hydrogen, oxygen, carbon-monoxide, and carbon-dioxide, 
and multiplying these respectively by the diffusivities from 
Loschmidt's experiments and Maxwell's theory, we have the 
following comparison with Obermayer 's viscosities at C. and 
standard pressure, which shows the discrepance from experiment 
and seeming theory referred to in 34. 



Col. 1 


Col. 2 


Col. 3 


Col. 4 


Gas 


Viscosity calculated 
by Maxwell's theory 
from Loschmidt's 
diffusivities 
H=pD 


Viscosities according 
to Obermayer 


liatio of values in 
Col. 3 to those 
in Col. 2 


H 2 


000119 


0000822 


691 


2 


000269 


0001873 


696 


CO 


000212 


0001630 


769 


C0 2 -000218 


0001414 


649 

I 



* The formula for viscosity (Sci. Papers, Vol. n. p. 68) taken with the formula 
for molecular diffusivity of a single gas, derived from the formula of mter-diffusivity 



MAXWELL'S THEORY. 297 

46. Leaving this discrepance unexplained, and eliminating Molecular. 
D between (1) of 44 and (3) of 41, we find as Maxwell's It test 
expression of the theoretical relation between number of molecules 
per cubic centimetre, diameter of the molecules, molecular velocity, 
density, and viscosity of a single gas, 

A*. .} ^ = -1629^ ..... (1). 

2V37T M /* 

The number of grammes and the number of molecules in a cubic 
centimetre being respectively p and N, p/N is the mass of one 
molecule in grammes ; and therefore, denoting this by m, we have 



* 2 = 6-140 * 2 .................. (2). 

In these formulas, as originally investigated by Maxwell for the 
case of an ideal gas composed of hard spherical atoms, s is 
definitely the diameter of the atom, and is the same at all 
temperatures and densities of the gas. When we apply the 
formulas to diatomic or polyatomic gases, or to a monatomic gas 
consisting of spherical atoms whose spheres of action may over- 
lap more or less in collision according to the severity of the 
impact, 8 may be defined as the diameter which an ideal hard 
spherical atom, equal in mass to the actual molecule, must have to 
give the same viscosity as the real gas, at any particular tem- 
perature. This being the rigorous definition of s, we may call it 
the proper mean shortest distance of inertial centres of the mole- 
cules in collision to give the true viscosity ; a name or expression 
which helps us to understand the thing defined. 

47. For the ideal gas of hard spherical atoms, remembering 
that V is independent of the density and varies as fi (t denoting 
absolute temperature), 46 (2) proves that the viscosity is inde- 
pendent of the density and varies approximately as t^. Rayleigh's 
experimental determinations of the viscosity of argon at different 
temperatures show that for this monatomic gas the viscosity varies 
as t m ; hence 46 (2) shows that s 2 varies as t~ M , and therefore 
* varies as t~' M . Experimental determinations by Obermayer* of 
viscosities and their rates of variation with temperature for car- 
bonic acid, ethylene, ethylene-chloride, and nitrous oxide, show 

of two gases of equal densities, gives 4; = '-f , which is equal to -648 according to 
pD 3A, 

the valnes of A l and A t shown in p. 42 of VoL n., Sci. Papert. 
* Obermayer, H"i*n. Akad. 1876, Mar. 16th, Vol. 73, p. 433. 



298 



LECTURE XVII. 



lolecular. that! for these the viscosity is somewhat nearly in simple propor- 
tion' to the absolute temperature : hence for them s- varies nearly 
as t~' 5 . His determinations for the five molecularly simpler gases, 
air, hydrogen, carbonic oxide, nitrogen, and oxygen show that the 
increases of /u, and therefore of s~ 2 , with temperature are, as 
might be expected, considerably smaller than for the more complex 
of the gases on which he experimented. Taking his viscosities 
at Cent., for carbonic acid and for the four other simple gases 
named above, and Rayleigh's for argon, with the known densities 
of all the six gases at C. and standard atmospheric pressure, 
we have the following table of the values concerned in 4u' (1): 









i 


j i 


Col. 1 


Col. 2 


Col. 3 


Col. 4 Col. 5 


Col. G 


Col. 7 Col. 8 Col. 9 












Mean free Hatio of 












paths ac- volume oc- 










Hence taking 


... ... conliiiK to cupivd by 
'/'K 111 .^ Maxwell's ; inoleciil. ^ 't. 








1 


^we'havV'"* w,."l!"v,: fnnula* whole volume 


(ias 





H V .\s- 


.? at 0" Cent. 






in ti-nns of 


in terms i" terms ot in terms ul 


in terms of in terms of ' v2.7rAVJ "6' 






of dynes centimetres (centimetre) ' 


centimetres 


grammes jn k . rnls of 




metre 


centimetre 


centimetres ' 


CO., 


001974 


0001414 , 39200 89200 


2-99 . 10- jl'J-74. 10- 24 2-52 . IQ-" i 1-31I0.10- 3 


H., 


00001)00 


0000822 184000 32800 


1-81 : -90 , 0-89 i -311 


CO 


001234 


0001030 49GOO G1200 


2-47 12-34 3-G8 : -792 ,, 


N 


001257 


0001035 49200 01GOO 


2-48 


12-57 3-GG ! -800 


0, 


00143 


0001873 4G100 57300 


2-39 14-3 3-93 -719 


Argon 


001781 


0002083 


41300 57500 


2-40 117-81 3-91 i -722 

I 



48. The meaning of "6-," the diameter, as defined in 46, is 
simpler for the monatomic gas, argon, than for any of the others ; 
and happily we know for argon the density, not only in the gaseous 
state (-001781) but also in the liquid state (l'42)f. The latter 
of these is 797 times the former. Now, all things considered, it 
seems probable that the crowd of atoms in the liquid may be 
slightly less dense than an assemblage of globes of diameter s just 
touching one another in cubic order ; but, to make no hypothesis 
in the first place, let qs be the distance from centre to centre 
of a cubic arrangement of the molecules 797 times denser than 
the gas at C. and standard atmospheric pressure ; q will be 
t,er than unity if the liquid is less dense, or less than unity 
C0_ 

well's Collected Papers, Vol. n. p. 348, eqn. (7). The formula as printed 
* The ft 1 contains a y ery embarrassing mistake, Jfrr for ^/2 . w. 
for molecular^ and Donan . Ckem. Jour., July, 1902. 



NUMBER OF MOLECULES IN A CUBIC CENTIMETRE OF GAo. 301 



if the liquid is denser, than the cubic arrangement with molecules, 
regarded as spherical of diameter 8, just touching. We have 

mN=ll(qsY (3), 

and for argon we have by 46 (1), 

N& = 57500 (4). 

Eliminating s between these equations we find 

#=797'. 57500V = 121. 10 s ". </ (5). 

If the atoms of argon were ideal hard globes, acting on one 
another with no force except at contact, we should almost cer- 
tainly have <? = 1 (because with closer packing than that of cubic 
order it seems not possible that the assemblage could have 
sufficient relative mobility of its parts to give it fluidity) and 
therefore N would be 1-21 . 10 

49. For carbonic acid, hydrogen, nitrogen, and oxygen, we 
have experimental determinations of their densities in the solid or 
liquid state ; and dealing with them as we have dealt with argon, 
irrespectively of their not being monatomic gases, we find results 
for the five gases as shown in the following table : 



Col. 1 


Col. 2 


Col. 3 


Col. 4 


Col. 5 


Gu 


Solid or liquid density 


Ratio of 
K.I id or 

liquid 
density to 

-l:iM.|.,i.l 
gaseous 
density 


Number of 
molecules per 
cubic centimetre 
of gas at stamlanl 
dendty 


Value, of ?) 
according to 
(j-=l -21 for argon 
OKI u id compared 
with KM at 0* and 
atmospheric 
pressure) 


CO, 


Solid ... .. 1-58 


800 


N 
4-55 .10.fl 


777 


H, 

N, 


liquid at 17 absolute ... -OIK) 
lliquid 1-047 
jsolid 1-400 


1000 
833 
1114 


352 T 
1-62 
2-90 


1-191 
923 
837 


o, 

Argon 


liquid at its freezing pt. 1-27 
( liquid at 84 absolute . . . 1-42 
solid 1-396* 


888 
797 
784 


1-49 
1-21 
1-17 


936 
969 
974 













In this table, q denotes the ratio to s of the distance from 
centre to centre of nearest molecules in an ideal cubic assembla?" 
of the same density as the solid or liquid, as indicated irdhow 

and 2. 

;cording to 

* From information communicated by Prof. W. Ramsay, July/ 



300 



LECTURE XVII. 



Tecular. 50. According to Avogadro's doctrine, the number of mole- 
cules per cubic centimetre is the same for all " perfect " gases at 
the same temperature and pressure ; and even carbonic acid is 
nearly enough a "perfect gas " for our present considerations. Hence 
the actual values of q 6 are inversely proportional to the numbers 
by which they are multiplied in col. 3 of the preceding table. 
Now, as said in 48, all things considered, it seems probable that 
for argon, liquid at density 1*42, q may be somewhat greater, but 
not much greater, than unity. If it were exactly unity, N would 
be 1'21. 10 20 ; and I have chosen </=(l-21)-* or "969, to make N 
the round number 10 ao . Col. 6, in the table of 47 above, is 
calculated with this value of N; but it is not improbable that the 
true value of N may be considerably greater than 10 20 *. 

51. As compared with the value for argon, monatomic, the 
smaller values of q for the diatomic gases, nitrogen and oxygen, 
and the still smaller values for carbonic acid, triatomic, are quite 
as might be expected without any special consideration of law of 
force at different distances between atoms. It seems that the 
diatomic molecules of nitrogen and oxygen and still more so the 
triatomic molecule of carbonic acid, are effectively larger when 
moving freely in the gaseous condition, than when closely packed 
in liquid or solid assemblage. But the largeness of q for the 
diatomic hydrogen is not so easily explained : and is a most in- 
teresting subject for molecular speculation, though it or any other 
truth in nature is to be explained by a proper law of force accord- 
ing to the Boscovichian doctrine which we all now accept (many 

* Maxwell, judging from "molecular volumes" of chemical elements estimated 
by Loreutz, Meyer and Kopp, unguided by what we now know of the densities 
of liquid oxygen and liquid hydrogen and of the liquid of the then undiscovered gas 
argon, estimated N=-l$ . 10 20 (Maxwell's Collected Papers, Vol. n. p. 350) which is 
rather less than one-fifth of my estimate 10 20 . On the same page of his paper 
is given a table of estimated diameters of molecules which are about 3-2 or 3-3 times 
larger than my estimates in col. 6 of the table in 47. In a previous part of 
his paper (p. 348) Maxwell gives estimates of free paths for the same gases, from 
which by his formula (7), corrected as in col. 8 of my table in 47, I find values of 
N ranging from 6-05 . 10 18 to 6-96 . 10 13 or about one-third of -19 . 10 20 . His 
uncorrected formula Jlir (instead of ^2 . IT) gives values of N which are *Jir times, 
or 1'77 times as great, which are still far short of his final estimate. The discrepance 
is therefore not accounted for by the error in the formula as printed, and I see no 
explanation of it. The free paths as given by Maxwell are about 1-3 or 1-4 times 
as large as mine. 



RAYLEIOH'S THEORY OF THE BLUE OF THE SKY. 301 

of us without knowing that we do so) as the fundamental hypo- Molecular 
thesis of physics and chemistry. I hope to return to this question 
as to hydrogen in a crystallographic appendix. 

I am deeply indebted to Professor Dewar for information 
regarding the density of liquid hydrogen, and the densities of 
other gases, liquefied or frozen, which he has given me at various 
times within the last three years. 

52. A new method of finding an inferior limit to the number 
of molecules in a cubic centimetre of a gas, very different from 
anything previously thought of, and especially interesting to us 
in connection with the wave-theory of light, was given by Lord 
Rayleigh*, in 1899, as a deduction from the dynamical theory of 
the blue sky which he had given 18 years earlier. Many previous 
writers, Newton included, had attributed the light from the sky, 
whether clear blue, or hazy, or cloudy, or rainy, to fine suspended 
particles which divert portions of the sunlight from its regular 
course; but no one before Rayleigh, so far as I know, had published 
any idea of how to explain the blueness of the cloudless sky. 
Stokes, in his celebrated paper on Fluorescence "f*, had given the 
true theory of what was known regarding the polarization of the 
blue sky in the following "significant remark" as Rayleigh calls it : 
" Now this result appears to me to have no remote bearing on the 
" question of the directions of the vibrations in polarized light. 
" So long as the suspended particles are large compared with the 
" waves of light, reflection takes place as it would from a portion of 
" the surface of a large solid immersed in the fluid, and no con- 
" elusion can be drawn either way. But if the diameter of the 
" particles be small compared with the length of a wave of light, it 
" seems plain that the vibrations in a reflected ray cannot be per- 
" pendicular to the vibrations in the incident ray"; which implies 
that the light scattered in directions perpendicular to the exciting 
incident ray has everywhere its vibrations perpendicular to the 
plane of the incident ray and the scattered ray ; provided the 
diameter of the molecule which causes the scattering is very small 
in comparison with the wave-length of the light. In conversation 
Stokes told me of this conclusion, and explained to me with 

* Rayleigh, Collected Papers, Vol. i. Art. vm. p. 87. 

t "On the Change of Refrangibility of Light," Phil. Trans. 1852, and Collected 
Paper*, Vol. in. 



302 LECTURE XVII. 

Molecular, perfect clearness and completeness its dynamical foundation ; 
and applied it to explain the polarization of the light of a cloud- 
less sky, viewed in a direction at right angles to the direction 
of the sun. But he did not tell me (though I have no doubt he 
knew it himself) why the light of the cloudless sky seen in any 
direction is blue, or I should certainly have remembered it. 

53. Rayleigh explained this thoroughly in his first paper 
(1871), and gave what is now known as Rayleigh's law of the blue 
sky; which is, that, provided the diameters of the suspended particles 
are small in comparison with the wave-lengths, the proportions of 
scattered light to incident light for different wave-lengths are 
inversely as the fourth powers of the wave-lengths. Thus, while 
the scattered light has the same colour as the incident light 
when homogeneous, the proportion of scattered light to incident 
light is seven times as great for the violet as for the red of the 
visible spectrum ; which explains the intensely blue or violet 
colour of the clearest blue sky. 

54. The dynamical theory shows that the part of the light 
of the blue sky, looked at in a direction perpendicular to the 
direction of the sun, which is due to sunlight incident on a single 
particle of diameter very small in comparison with the wave- 
lengths of the illuminating light, consists of vibrations perpen- 
dicular to the plane of these two directions : that is to say, is 
completely polarized in the plane through the sun. In his 1871 
paper*, Rayleigh pointed out that each particle is illuminated, not 
only by the direct light of the sun, but also by light scattered 
from other particles, and by earth-shine, and partly also by sus- 
pended particles of dimensions not small in comparison with the 
wave-lengths of the actual light ; and he thus explained the 
observed fact that the polarization of even the clearest blue sky 
at 90 from the sun is not absolutely complete, though it is very 
nearly so. There is very little of polarization in the light from 
white clouds seen in any direction, or even from a cloudless sky 
close above the horizon seen at 90 from the sun. This is partly 
because the particles which give it are not small in comparison 
with the wave-lengths, and partly because they contribute much 
to illuminate one another in addition to the sunlight directly 
incident on them. 

* Collected Papers, Vol. i. p. 94. 



RAYLEIGH'S THEORY OF THE BLUE OF THE SKY. 303 

55. For his dynamical foundation, Rayleigh definitely MolecnU 
assumed the suspended particles to act as if the ether in their 
places were denser than undisturbed ether, but otherwise unin- 
fluenced by the matter of the particles themselves. He tacitly 
assumed throughout that the distance from particle to particle 
is very great in comparison with the greatest diameter of each 
particle. He assumed these denser portions of ether to be of the 
same rigidity as undisturbed ether; but it is obvious that this 
last assumption could not largely influence the result, provided 
the greatest diameter of each particle is very small in comparison 
with its distance from next neighl>our, and with the wave-lengths 
of the light : and, in fact, I have found from the investigation of 
41, 42 of Lecture XIV. for rigid spherical molecules embedded 
in ether, exactly the same result as Rayleigh 's ; which is as follows 

8vn //)' - D 2V ID' - D 2V 

*- T\TTT vj =82>67w ( 7rv) ...... (1); 

where \ denotes the wave-length of the incident light supposed 
homogeneous ; T the volume of each suspended particle ; D the 
undisturbed density of the ether; If the mean density of the 
ether within the particle ; n the number of particles per cubic 
centimetre; and k the proportionate lass of homogeneous incident 
light, due to the scattering in all directions by the suspended 
particles per centimetre of air traversed. Thus 



is the loss of light in travelling a distance x (reckoned in centi- 
metres) through ether as disturbed by the suspended particles. 

It is remarkable that D' need not be uniform throughout the 
particle. It is also remarkable that the shape of the volume T 
may be anything, provided only its greatest diameter is very 
small in comparison with X. The formula supposes T \H D) 
the same for all the particles. We shall have to consider cases in 
v'hich differences of T and D' for different particles are essential to 
the result ; and to include these we shall have to use the formula 



where 2 [^ -/>>*]' denote, the sum of [^ >/> Z ']' for all 
the particles in a cubic centimetre. 



304 LECTURE XVII. 

Molecular. 56. Supposing now the number of suspended particles per 
cubic wave-length to be very great, and the greatest diameter 
of each to be small in comparison with its distance from next 
neighbour, we see that the virtual density of the ether vibrating 
among the particles is 

D) ........................ (4); 



and therefore, if u and u be the velocities of light in pure ether, 
and in ether as disturbed by the suspended particles, we have 
(Lecture VIII. p. 80) 



Hence, if /A denote the refractive index of the disturbed ether 
that of pure ether being 1, we have 



and therefore, 



57. In taking an example to illustrate the actual trans- 
parency of our atmosphere, Rayleigh says* ; " Perhaps the best 
" data for a comparison are those afforded by the varying bright- 
" ness of stars at various altitudes. Bouguer and others esti- 
" mate about '8 for the transmission of light through the entire 
" atmosphere from a star in the zenith. This corresponds to 8*3 
" kilometres (the " height of the homogeneous atmosphere " at 
" 10 Cent.) of air at standard pressure." Hence for a medium of 
the transparency thus indicated we have e~ 830000fc = '8 ; which gives 
I/A; = 3720000 centimetres = 37'2 kilometres. 

58. Suppose for a moment the want of perfect trans- 
parency thus denned to be wholly due to the fact that the 
ultimate molecules of air are not infinitely small and infinitely 
numerous, so that the " suspended particles " hitherto spoken of 
would be merely the molecules N 2 , O 2 ; and suppose further 
(D'D)T to be the same for nitrogen and oxygen. The known 

* Phil. Mag. April, 1899, p. 382. 



NUMBER OF MOLECULES PER CUBIC CENTIMETRE OF GAS. 305 

refractivity of air (p, 1 = '0003), nearly enough the same for Molecular. 
all visible light, gives by equation (7) above, with n instead of S, 



Using this in (1) we find 



29-76 

- 



for what the rate of loss on direct sunlight would be, per centi- 
metre of air traversed, if the light were all of one wave-length, X. 
But we have no such simplicity in Bouguer's datum regarding 
transparency for the actual mixture which constitutes sunlight : 
because the formula makes kr l proportional to the fourth power 
of the wave-length ; and every cloudless sunset and moonset and 
sunrise and moonrise over the sea, and every cloudless view of 
sun or moon below the horizon of the eye on a high mountain, 
proves the transparency to be in reality much greater for red 
light than for the average undimmed light of either luminary, 
though probably not so much greater as to be proportional to the 
fourth power of the wave-length. We may, however, feel fairly 
sure that Bouguer's estimate of the loss of light in passing 
vertically through the whole atmosphere is approximately true 
for the most luminous part of the spectrum corresponding to 
about the D line, wave-length 5*89 . 10~ 8 cm., or (a convenient 
round number) 6 . 10~ 5 as Rayleigh has taken it. With this value 
for X, and 372 . 10 6 centimetres for k~\ (8) gives n = 8-54 . 10 18 for 
atmospheric air at 10 and at standard pressure. Now it is quite 
certain that a very large part of the loss of light estimated by 
Bouguer is due to suspended particles ; and therefore it is certain 
that the number of molecules in a cubic centimetre of gas, at 
standard temperature and pressure, is considerably greater than 
8-54 . 10 18 . 

59. This conclusion drawn by Rayleigh from his dynamical 
theory of the absorption of light from direct rays through air, 
giving very decidedly an inferior limit to the number of molecules 
in a cubic centimetre of gas, is perhaps the most thoroughly well 
founded of all definite estimates hitherto made regarding sizes or 
numbers of atoms. We shall see ( 73... 79, below) that a much 
larger inferior limit is found on the same principles by careful 
T. L. 20 



306 LECTURE XVII. 

Molecular, consideration of the loss of light due to the ultimate molecules 
of pure air and to suspended matter undoubtedly existing in all 
parts of our atmosphere, even where absolutely cloudless, that is 
to say, warmer than the dew-point, and therefore having none of 
the liquid spherules of water which constitute cloud or mist. 

60. Go now to the opposite extreme from the tentative 
hypothesis ot 58, and, while assuming, as we know to be true, 
that the observed refractivity is wholly or almost wholly due to 
the ultimate molecules of air, suppose the opacity estimated by 
Bouguer to be wholly due to suspended particles which, for 
brevity, we shall call dust (whether dry or moist). These par- 
ticles may be supposed to be generally of very unequal magni- 
tudes : but, for simplicity, let us take a case in which they are all 
equal, and their number only 1/10000 of the 8'54.10 18 , which 
in 59 we found to give the true refractivity of air, with 
Bouguer's degree of opacity for X = 6.10~ 5 . With the same 
opacity we now find the contribution to refractivity of the 
particles causing it, to be only 1/100 of the known refractivity 
of air. The number of particles of dust which we now have is 
(S'54.10 14 per cubic centimetre, or 184 per cubic wave-length, 
which we may suppose to be almost large enough or quite large 
enough to allow the dynamics of 56 for refractivity to be 
approximately true. But it seems to me almost certain that 
8-54. 10" is vastly greater than the greatest number of dust 
particles per cubic centimetre to which the well-known haziness 
of the clearest of cloudless air in the lower regions of our 
atmosphere is due ; and that the true numbers, at different times 
and places, may probably be such as those counted by Aitken* 
at from 42500 (Hyeres, 4 p.m. April 5, 1892) to 43 (Kingairloch, 
Argyllshire, 1 p.m. to 1.30 p.m. July 26, 1891). 

61. Let us, however, find how small the number of par- 
ticles per cubic centimetre must be to produce Bouguer's degree 
of opacity, without the particles themselves being so large in 
comparison with the wave-length as to exclude the application of 
Rayleigh's theory. Try for example T= 10" 3 . X 3 (that is to say, 
the volume of the molecule 1/1000 of the cubic wave-length, or 

* Tram. P. S. E. 1894, Vol. xxxvn. Part m. pp. 675. (572. 



HAZE DUE TO PARTICLES TOO LARGE FOR RAYLEIGH'S THEORY. 307 

roughly, diameter of molecule 1/10 of the wave-length) which Molecular, 
seems small enough for fairly approximate application of Ray- 
leigh's theory; and suppose, merely to make an example, H to be 
the optical density of water, D being that of ether ; that is to say, 
UID = (1-3337)' = 1-78. Thus we have (D' - D) TfD = -00078X': 
and with X=6.10~ 8 , and with fc-^3'72 . 10,(1) gives n= 1'485.10, 
or about one and a half million particles per cubic centimetre. 
Though this is larger than the largest number counted for natural 
air by Aitken, it is interesting as showing that Bouguer's degree of 
opacity can be accounted for by suspended particles, few enough 
to give no appreciable contribution to refractivity, and yet not too 
large for Rayleigh's theory. But when we look through very 
clear air by day, and see how far from azure or deep blue is the 
colour of a few hundred metres, or a few kilometres of air with 
the mouth of a cave or the darkest shade of mountain or forest, 
for background ; and when in fine sunny weather we study the 
appearance of the grayish haze always, even on the clearest days, 
notably visible over the scenery among mountains or hills ; and 
when by night at sea we see a lighthouse light at a distance of 
45 or 50 kilometres, and perceive how little of redness it shows ; 
and when we see the setting sun shorn of his brilliance sufficiently 
to allow us to look direct at his face, sometimes whitish, oftener 
ruddy, rarely what could be called ruby red ; it seems to me that 
we have strong evidence for believing that the want of perfect 
clearness of the lower regions of our atmosphere is in the main 
due to suspended particles, too large to allow approximate ful- 
filment of Rayleigh's law of fourth power of wave-length. 

62. But even if they were small enough for Rayleigh's 
theory the question would remain, Are they small enough and 
numerous enough to account for the refractivity of the atmo- 
sphere ? To this we shall presently see we must answer un- 
doubtedly "No"; and much less than Bouguer's degree of opacity, 
probably not as much as a quarter or a fifth of it, is due to the 
ultimate molecules of air. In a paper by Mr Quirino Majorana in 
the Transactions of the R. Accademia dei Lincei (of which a 
translation is published in the Philosophical Magazine for May, 
1901), observations by himself in Sicily, at Catania and on Mount 
Etna, and by Mr Gaudenzio Sella, on Monte Rosa in Switzerland, 
determining the ratio of the brightness of the sun's surface to the 

202 



308 LECTURE XVII. 

Molecular, brightness of the sky seen in any direction, are described. This 
ratio they denote by r. One specially notable result of Mr 
Majorana's is that " the value of r at the crater of Etna is about 
five times greater than at Catania." The barometric pressures 
were approximately 53'6 and 76 cms. of mercury. Thus the atmo- 
sphere above Catania was only 1'42 times the atmosphere above 
Etna, and yet it gave five times as much scattering of light by its 
particles, and by the particles suspended in it. This at once 
proves that a great part of the scattering must be due to sus- 
pended particles ; and more of them than in proportion to the 
density in the air below the level of Etna than in the air above it. 
In Majorana's observations, it was found that " except for regions 
" close to the horizon, the luminosity of the sky had a sensibly 
" constant value in all directions when viewed from the summit of 
" Etna." This uniformity was observed even for points in the 
neighbourhood of the sun, as near to it as he could make the 
observation without direct light from the sun getting into his 
instrument. I cannot but think that this apparent uniformity 
was only partial. It is quite certain that with sunlight shining 
down from above, and with light everywhere shining up from 
earth or sea or haze, illuminating the higher air, the intensities 
of the blue light seen in different directions above the crater 
would be largely different. This is proved by the following 
investigation ; which is merely an application of Rayleigh's theory 
to the question before us. But from Majorana's narrative we 
may at all events assume that, as when observing from Catania, 
he also on Etna chose the least luminous part of the sky (Phil. 
Mag., May 1901, p. 561), for the recorded results (p. 562) of his 
observations. 

63. The diagram, fig. 1 below, is an ideal representation of 
a single molecule or particle, T, with sunlight falling on it indi- 
cated by parallel lines, and so giving rise to scattered light seen 
by an eye at E. We suppose the molecule or particle to be so 
massive relatively to its bulk of ether that it is practically un- 
moved by the ethereal vibration ; and for simplicity at present we 
suppose the ether to move freely through the volume T, becoming 
effectively denser without changing its velocity when it enters 
this fixed volume, and less dense when it leaves. In 41, 42, of 
Lecture XV. above, and in Appendix A, a definite supposition, 



DYNAMICS OF BLUE SKY. 309 

attributing to ether no other -property than elasticity as of an Molecular, 
utterly homogeneous perfectly elastic solid, and the exercise of 
mutual force between itself and ponderable matter occupying 
the same space, is explained: according to which the ether within 




Fig. i. 



the atom will react upon moving ether outside, just as it would 
if our present convenient temporary supposition of magically 
augmented density within the volume of an absolutely fixed 
molecule were realized in nature. For our present purpose, we 
may if we please, following Rayleigh, do away altogether with 
the ponderable molecule, and merely suppose T to be not fixed, 
but merely a denser portion of the ether. And if its greatest 
diameter is small enough relatively to a wave-length, it will 
make no unnegligible difference whether we suppose the ether in 
T to have the same rigidity as the surrounding free ether, or 
suppose it perfectly rigid as in 1 46 of Lecture XIV. dealing 
with a rigid globe embedded in ether. 

b'4. Resolving the incident light into two components having 
semi-ranges of vibration vr, p, in the plane of the paper and 
perpendicular to it ; consider first the component in the plane 



.310 LECTURE XVII. 

Molecular, having vibrations symbolically indicated by the arrow-heads, and 
expressed by the following formula 



where u is the velocity of light, and A, the wave-length. The 
greater density of the ether within T gives a reactive force on 
the surrounding ether outside, in the line of the primary vibration, 
and against the direction of its acceleration, of which the magni- 
tude is 



This alternating force produces a train of spherical waves spread- 
ing out from T in all directions, of which the displacement is, at 
greatest, very small in comparison with tj ; and which at any 
point E at distance r from the centre of T, large in comparison 
with the greatest diameter of T, is given by the following ex- 
pression* 

cos (ut r), 

A, 

with = --^.'~ D) cos 6 .................. (10), 

V'A* J-s 

where 6 is the angle between the direction of the sun and the 
line TE. Formula (10), properly modified to apply it to the other 
component of the primary vibration, that is, the component per- 
pendicular to the plane of the paper, gives for the displacement at 
E due to this component 

77 cos " (ut r), 



Hence for the quantity of light falling from T per unit of time, 

* This formula is readily found from 41, 42 of Lecture XIV. The complexity 
of the formulas in 8 40 is due to the inclusion in the investigation of forces and 
displacements at small distances from T, and to the condition imposed that T 
is a rigid spherical figure. The dynamics of 3336 with c = 0, and the details of 
3739 further simplified by taking v = co, lead readily to the formulas (10) and 
(11) in our present text. 



BRIGHTNESS OF BLUE SKY. 311 

on unit area of a plane at E, perpendicular to ET, reckoned in Moleculai 
convenient temporary units, we have 



65. Consider now the scattered light emanating from a large 
horizontal plane stratum of air 1 cm. thick. Let T of fig. 1 be one 
of a vast number of particles in a portion of this stratum sub- 
tending a small solid angle ft viewed at an angular distance ft 
from the zenith by an eye at distance r. The volume of this 
portion of the stratum is ft sec ft r 3 cubic centimetres ; and there- 
fore, if 2 denotes summation for all the particles in a cubic centi- 
metre, small enough for application of Rayleigh's theory, and 
q the quantity of light shed by them from the portion ft sec /8 r 3 
of the stratum, and incident on a square centimetre at E, per- 
pendicular to ET, we have 



Summing this expression for the contributions by all the 
luminous elements of the sun and taking 



to denote this summation, we Tiave instead of the factor 
r 3 cos 3 6 + p*, 

cos s 6 f-or 2 + tp- : 

and we have [ 1ff *=jp* = S ........................ (14), 

where S denotes the total quantity of light from the sun falling 
perpendicularly on unit of area in the particular place of the 
atmosphere considered. Hence the summation of (13) for all the 
sunlight incident on the portion ft sec ft i a of the stratum, gives 



66. To define the point of the sky of which the illumination 
is thus expressed, let f be the zenith distance of the sun, and ^r 
the azimuth, reckoned from the sun, of the place of the sky seen 



312 



LECTURE XVII. 



Molecular, along the line ET. This place and the sun and the zenith are at 
the angles of a spherical triangle SZT, of which ST is equal to 6. 
Hence we have 

cos 6 = cos cos ft + sin sin /3 cos ^ ............ (16). 

Let now, as an example, the sun be vertical: we have =0, 
6 = /3, and (15) becomes 



(17). 



This shows least luminosity of the sky around the sun at the 
zenith, increasing to oo at the horizon (easily interpreted). The 
law of increase is illustrated in the following table of values of 
(cos /3 + sec /3) for every 10 of /8 from to 90. 



, JHco^sec,, 


! (3 


<, + .. 





1-000 


50 


1-099 


10 


1-000 


60 


1-250 


20 


1-002 


70 1-633 


30" 


1-010 


80 2-966 


40 


1-036 


90 


GO 



67. Instead now of considering illumination on a plane 
perpendicular to the line of vision, consider the illumination by 
light from our one-centimetre-thick great* horizontal plane 
stratum of air, incident on a square centimetre of horizontal 
plane. The quantity of this light per unit of time coming from 
a portion of sky subtending a small solid angle fl at zenith 
distance /3 is Qcos/3. Taking fl = sin /3 d(3 d-ty- and integrating, 
we find for the light shed by the one-centimetre-thick horizontal 
stratum on a horizontal square centimetre of the ground, 



Now each molecule and particle of dust sheds as much light 
upwards as downwards. Hence (18) doubled expresses the 

* We are neglecting the curvature of the earth, and supposing the density and 
composition of the air to be the same throughout the plane horizontal stratum to 
distances from the zenith very great in comparison with its height above the 
ground. 



SKY- LIGHT OVER ETNA AND MONTE ROSA. 313 

quantity of light lost by direct rays from a vertical sun in cross- Molecular 
ing the one-centimetre-thick horizontal stratum. It agrees with 
the expression for k in (1) of 55, as it ought to do. 

68. The expression (15) is independent of the distance of 
the stratum above the level of the observer's eye. Hence if H 
denote the height above this level, of the upper boundary of an 
ideal homogeneous atmosphere consisting of all the ultimate 
molecules and all the dust of the real atmosphere scattered uni- 
formly through it, and if 8 denote the whole light on unit area of 
a plane at E perpendicular to ET, from all the molecules and dust 
in the solid angle H of the real atmosphere due to the sun's direct 
light incident on them, we have 

i = *^2[^>]'n.i<cos* + l) (19); 

provided we may, in the cases of application whatever they 
may be, neglect the diminution of the direct sunlight in its 
actual course through air, whether to the observer or to the 
portion of the air of which he observes the luminosity, and neglect 
the diminution of the scattered light from the air in its course 
through air to the observer. This proviso we shall see is prac- 
tically fulfilled in Mr Majorana's observations on the crater of 
Etna for zenith distances of the sun not exceeding 60, and in 
Mr Sella's observation on Monte Rosa in which the sun's zenith 
distance was 50. But for Majorana's recorded observation on 
Etna at 5.50 a.m. when the sun's zenith distance was 81'71, of 
which the secant is 6'928, there may have been an important 
diminution of the sun's light reaching the air vertically above the 
observer, and a considerably more important diminution of his 
light as seen direct by the observer. This would tend to make 
the sunlight reaching the observer less strong relatively to the 
sky-light than according to (19); and might conceivably account 
for the first number in col. 8 being smaller than the first number 
in col. 4 of the Table of 69 below ; but it seems to me more 
probable that the smallness of the first two numbers in col. 3, 
showing considerably greater luminosity of sky than according to 
(19), may be partly or chiefly due to dust in the air overhead, 
optically swelled by moisture in the early morning. Indeed the 
largeness of the luminosity of the sky indicated by the smallness 



314 



LECTURE XVII. 



violecular. of the first three numbers in col. 3, in comparison with the corre- 
sponding numbers of col. 4, is explained most probably by gray 
haze in the early morning. 

69. The results of Majorana's observations from the crater 
of Etna are shown in the following Table, of which the first and 
third columns are quoted from the Philosophical Magazine for 
May, 1901, and the second column has been kindly given to me 
in a letter by Mr Majorana. The values of S/s shown in col. 4 
are calculated from 68 (19), with the factor of sec /3 (cos 2 9 + 1) 
taken to make it equal to Majorana's r for sun's zenith distance 
44'6, on the supposition that the region of sky observed was in 
each case (see 62 above) in the position of minimum luminosity 
as given by (21). It is obvious that this position is in a vertical 



Col. 1 


Col. 2 


Col. 3 


Col. 4 


Col. 5 


Time 


Zenith 
distance 
of sun. 
I 


Ratio of 
luminosity of 
sun's disc to 
luminosity of 
sky. 
r 


AS' 

8 


Zenith distance 
of least lumi- 
nous part of 

J 


5.50 A.M. 


81 ! 7 


2570000 


4820000 


5 ? 5 


7 


68-0 


3125000 


4610000 


14-4 


8 


56-1 


3650000 


4310000 


21-7 


9 


44-6 


3930000 


3930000 


27-8 


11 


29-9 


3760000 


3370000 


33-6 



great circle through the sun, and on the opposite side of the 
zenith from the sun; and thus we have #=+$. Hence (19) 
becomes 



To make (20) a minimum we have 



The value of /3 satisfying this equation for any given value of 
is easily found by trial and error, guided by a short preliminary 



SKY-LIGHT OVER ETNA. 315 

table of values of $ for assumed values of ft -t . Col. 5 shows Molecula 
values of ft thus found approximately enough to give the values of 
S/s shown in col. 4 for the several values of f. 

70. Confining our attention now to Majorana's observations 
at 9 a.m. when the sun's altitude was about 44 0< 6 ; let e be the 
proportion of the light illuminating the air over the crater of 
Etna which at that hour was due to air, earth, and water below ; 
and therefore 1 e the proportion of the observed luminosity of 
the sky which was due to the direct rays of the sun, and expressed 
by 68 (19). Thus, for ft = 27'8, = 44-6, and = 72'4, we have 
S/s = 3930000/(1 - e), instead of the S/s of col. 4, 69. With this, 
equation (20) gives 



(22) 



Here, in order that the comparison may be between the whole 
light of the sun and the light from an equal apparent area of the 
sky, we must take 

fl = 7r/214-6* = 1/14660, 

being the apparent area of the sun's disc as seen from the earth. 
As to H, it is what is commonly called the " height of the homo- 
geneous atmosphere" and, whether at the top of Etna or at 
sea-level, is 

7-988. 10" (l + ~) centimetres; 

where t denotes the temperature at the place above which U is 
reckoned. Taking this temperature as 15 C., we find 

H = 8-43 . 10 5 centimetres. 
Thus (22) becomes 



v \ T (&- ^T = x (1 - e) . "728 . 10- ......... (23). 

71. Let us now denote by /and 1 / the proportions of 
(23) due respectively to the ultimate molecules of air and to 
dust. We have 



....... (24); 

* The sun's distance from the earth is 214-6 times his radius. 



316 LECTURE XVII. 

Secular, where n denotes the number of the ultimate molecules in a cubic 
centimetre of the air at the top of Etna ; and T (D r D)/D relates 
to any one of these molecules ; any difference which there may be 
between oxygen and nitrogen being neglected. Now assuming 
that the refractivity of the atmosphere is practically due to the 
ultimate molecules, and that no appreciable part of it is due to the 
dust in the air, we have by 56 (7), 



the first number being approximately enough the refractivity of 
air at the crater of Etna (barometric pressure, 53'6 centimetres of 
mercury). Hence 



(26,, 



and using this in (24) we find 

"- x ^-) <> 

Here, as in 58 in connection with Bouguer's estimate for loss 
of light in transmission through air, we have an essential un- 
certainty in respect to the effective wave-length ; and, for the 
same reasons as in 58, we shall take X = 6 . 10~ 5 cm. as the 
proper mean for the circumstances under consideration. With 
this value of X, (27) becomes 



72. In Mr Sella's observations on Monte Rosa the zenith 
distance of the sun was 50, and the place of the sky observed was 
in the zenith. He found the brightness of the sun's disc to be 
about 5000000 times the brightness of the sky in the zenith. 
Dealing with this result as in 70, 71, with /3=0 in (20), 
and supposing the temperature of the air at the place of obser- 
vation to have been C., we find 

/ 



where e', /', and n are the values of e, /, and n, at the place of 
observation on Monte Rosa. Denoting now by N the number of 
molecules in a cubic centimetre of air at C. and pressure 



SKY- LIGHT OVER MONTE ROSA. 



317 



75 centimetres of mercury, we have, by the laws of Boyle and Molecula 
Charles, on the supposition that the temperature of the air was 
1 5 on the summit of Etna, and on Monte Rosa 



49' 



or N= l-48/i = l'53n' 

From these, with (28) and (29), we find 



.(30). 



73. To estimate the values of e and e' as defined in 70, 72, 
consider the albedos* of the earth as might be seen from a balloon 
in the blue sky observed by Majorana and G. Sella over Etna and 
over Monte Rosa respectively. These might be about '2 and '4, 
the latter much the greater because of the great amount of snow 
contributing to illuminate the sky over Monte Rosa. With so 
much of guess-work in our data we need not enter on the full 
theory of the contribution to sky-light by earth-shine from below 
according to the principle of 67, G8, interesting as it is ; and we 
may take as very rough estimates *2 and *4 as the values of e and 
e'. Thus (3 1 ) becomes 



.(32). 



74. Now it would only be if the whole light of the sky were 
due to the ultimate molecules on which the refractivity depends 
that f or f could have so great a value as unity. If this were 
the case for the blue sky seen over Monte Rosa by G. Sella in 

* Albedo is a word introduced by Lambert 150 years ago to signify the ratio of 
the total light emitted by a thoroughly unpolished solid or a mass of cloud to the 
total amount of the incident light. The albedo of an ideal perfectly white body 
is 1. My friend Professor Becker has kindly given me the following table of albedos 
from Miiller's book Die Photometric der Oeitirne (Leipsic, 1897) as determined by 
observers and experimenters. 



Mercury 0-14 


Uranus 0-60 


Venus 0-76 


Neptune 0-52 


Moon 0-34 


Snow 0-78 


Mars 0-22 


White Paper 0-70 


Jupiter 0-62 


White Sandstone 0-24 


Saturn 0-72 


Damp Soil 0-08 



318 LECTURE XVII. 

lecular. 1900, we should have /' = 1 , and therefore N = 5*97 . 10 19 . But it 
is most probable that even in the very clearest weather on the 
highest mountain, a considerable portion of the light of the sky 
is due to suspended particles much larger than the ultimate 
molecules N 2 , O 2 , of the atmosphere ; and therefore the observa- 
tions of the luminosity of the sky over Monte Rosa in the summer 
of 1900 render it probable that N is greater than 5 '97 . 10 19 . If 
now we take our estimate of 50, for the number of molecules in 
a cubic cm. of air at 0, and normal pressure, N = 10 20 , we have 
1 -/= -088 and 1 -/' = '403 ; that is to say, according to the 
several assumptions we have made, '088 of the whole light of the 
portion of sky observed over Etna by Majorana was due to dust, 
and only '403 of that observed by Sella on Monte Rosa was due 
to dust. It is quite possible that this conclusion might be exactly 
true, and it is fairly probable that it is an approximation to the 
truth. But on the whole these observations indicate, so far as 
they can be trusted, the probability of at least as large a value as 
10 20 for N. 

S 75. All the observations referred to in 8s 57 74 are vitiated 

00 

by essentially involving the physiological judgment of perception 
of difference of strengths of two lights of different colours. In 
looking at two very differently tinted shadows of a pencil side by 
side, one of them blue or violet cast by a comparatively near 
candle, the other reddish-yellow cast by a distant brilliantly white 
incandescent lamp or by a more distant electric arc-lamp, or by 
the moon ; when practising Rum ford's method of photometry; it is 
quite wonderful to find how unanimous half-a-dozen laboratory 
students, or even less skilled observers, are in declaring This is 
the stronger ! or, That is the stronger ! or, Neither is stronger 
than the other ! When the two shadows are declared equally 
strong, the declaration is that the differently tinted lights from 
the two shadowed places side by side on the white paper are, 
according to the physiological perception by the eye, equally 
strong. But this has no meaning in respect to any definite 
component parts of the two lights; and the unanimity, or the 
greatness of the majority, of the observers declaring it, only proves 
a physiological agreement in the perceptivity of healthy average 
eyes (from which colour-blind eyes would no doubt differ wildly). 
Two circular areas of white paper in Sella's observations on 



PRISMATIC ANALYSIS OF SKY-LIGHT. 319 

Monte Rosa, a circle and a surrounding area of ground glass in Molecular. 
Majorana'a observations with his own beautiful sky-photometer 
on Etna, are seen illuminated respectively by diminished sunlight 
of unchanged tint and by light from the blue sky. The sun-lit 
areas seem reddish-yellow by contrast with the sky-lit areas which 
are azure blue. What is meant when the two areas differing so 
splendidly are declared to be equally luminous? The nearest 
approach to an answer to this question is given at the end of 
71 above, and is eminently unsatisfactory. The same may be 
truly said of the dealing with Bouguer's datum in 57, though 
the observers on whom Bouguer founded do not seem to have 
been disturbed by knowledge that there was anything indefinite 
in what they were trying to define or to find by observation. 

76. To obtain results not vitiated by the imperfection of 
the physiological judgment described in 75, Newton's prismatic 
analysis of the light observed, or something equivalent to it, is 
necessary. Prismatic analysis was used by Rayleigh himself for 
the blue light of the sky, actually before he had worked out 
his dynamical theory. He compared the prismatic spectrum of 
light from the zenith with that of sunlight diffused through 
white paper; and by aid of a curve drawn from about thirty 
comparisons ranging over the spectrum from C to beyond F, 
found the following results for four different wave-lengths. 

C D b a F 

Wave-length 656'2 589'2 517'3 48G'2 

Observed brightness ... 1 T64 2'84 3'60 

Calculated according to \~* . 1 1'54 2'52 3'34 

On these he makes the following remarks : " It should be 
"noticed that the sky compared with diffused light was even 
" bluer than theory makes it, on the supposition that the diffused 
" light through the paper may be taken as similar to that whose 
" scattering illuminates the sky. It is possible that the paper 
" was slightly yellow ; or the cause may lie in the yellowness 
" of sunlight as it reaches us compared with the colour it 
" possesses in the upper regions of the atmosphere. It would 
"be a mistake to lay any great stress on the observations in 
" their present incomplete form ; but at any rate they show that 
"a colour more or less like that of the sky would result from 



320 LECTURE XVII. 

>lecular. " taking the elements of white light in quantities proportional 
" to X~ 4 . I do not know how it may strike others ; but in- 
dividually I was not prepared for so great a difference as the 
" observations show, the ratio for F being more than three times 
"as great as for C." For myself I thoroughly agree with this 
last sentence of Rayleigh's. There can be no doubt of the 
trustworthiness of his observational results ; but it seems to me 
most probable, or almost certain, that the yellowness, or orange- 
colour, of the sunlight seen through the paper, caused by larger 
absorption of green, blue, and violet rays, explains the extreme 
relative richness in green blue, and violet rays which the results 
show for the zenith blue sky observed. 

77. An elaborate series of researches on the blue of the 
sky on twenty-two days from July, 1900, to February, 1901, is 
described in a very interesting paper, " Ricerche sul Bleu del 
Cielo," a dissertation presented to the "Royal University of Rome 
by Dr Giuseppe Zettwuch, as a thesis for his degree of Doctor 
in Physics. In these researches, prismatically analysed light from 
the sky was compared with prismatically analysed direct sunlight 
reduced by passage through a narrow slit ; and the results were 
therefore not vitiated by unequal absorptions of direct sunlight 
in the apparatus. A translation of the author's own account 
of his conclusions is published in the Philosophical Magazine 
for August, 1902; by which it will be seen that the blueness 
of the sky, even when of most serene azure, was always much 
less deep than the true Rayleigh blue denned by the \~ 4 law. 
Hence, according to Rayleigh's theory (see 53 above) much of 
the light must always have come from particles not exceedingly 
small in proportion to the wave-length. Thus in Zettwuch's 
researches we have a large confirmation of the views expressed 
in 54, 58, 61, 74 above, and 78, 79 below. 

78. Through the kindness of Professor Becker, I am now 
able to supplement Bouguer's 170-year old information with the 
results of an admirable extension of his investigation by Professor 
Mliller of the Potsdam Observatory, in which the proportion 
(denoted by p in the formula below) transmitted down to sea-level 
of homogeneous light entering our atmosphere vertically is found 
for all wave-lengths from 4'4.10~ s to 6'8.1()- 5 , by comparison of 



ATMOSPHERIC ABSORPTION ANALYSED BY HOLLER AND BECKER. 321 

the solar spectrum with the spectrum of a petroleum flame, for Molecular, 
different zenith distances of the sun. It is to be presumed, 
although I do not find it so stated, that only the clearest 
atmosphere available at Potsdam was used in these observations. 
For the sake of comparison with Rayleigh's theory, Professor 
Becker has arithmetically resolved Miiller's logarithmic results 
into two parts ; one constant, and the other varying inversely as 
the fourth power of the wave-length. The resulting formula*, 
modified to facilitate comparison with 57 59 above, is as 

follows : 

p = -(-+wr-) = .9 1 52 e --W7- (33), 

where z = X -r- 6 . 10~ 6 . In respect to the two factors here shown, 
we may say roughly that the first factor is due to suspended 
particles too large, and the second to particles not too large, for 
the application of Rayleigh's law. For the case of X = 6 . 10~ 8 
(*= 1) this gives 

p = -9152 . -9258 = '847 (34). 

79. Taking now the last term in the index and the last 
factor shown in (34) and dealing with it according to 57 59 
above; and still, as in 55, using k to denote the proportionate 
loss of light per centimetre due to particles small enough for 
Rayleigh's theory, whether " suspended particles " or ultimate 
molecules of air, or both ; we have "****** = '9258 which gives 
Ar 1 = 10-76 . 10* cms. Hence if, as in 58, we suppose for a 
moment the want of perfect transparency thus defined to be 
wholly due to the ultimate molecules of air, we should have, by 

the dynamics of refractivity, n ^ ~ J - ' = '0006 ; and thence by 

(1) of 55 with X = 6 . 10- we should find for the number of 
molecules per cubic centimetre n = 2'47 . 10 10 . But it is quite 
certain that a part, and most probable that a large part, of the want 
of transparency produced by particles small enough for Rayleigh's 
theory is due to " suspended particles " larger than the ultimate 
molecules: and we infer that the number of ultimate molecules 
per cubic centimetre is greater than, and probably very much 
greater than, 2 47 . 10 1 '. Thus from the surer and more complete 
data of Miiller regarding extinction of light of different wave- 

Miiller, Die Photometric der Gettirne, p. 140. 
T. L. 21 



322 



LECTURE XVII. 



Molecular, lengths traversing the air, we find an inferior limit for the number 
of molecules per cubic centimetre nearly three times as great as 
that which Rayleigh showed to be proved from Bouguer's datum. 

80. Taking, somewhat arbitrarily, as the result of 2377 
that the number of molecules in a cubic centimetre of a perfect 
gas at standard temperature and pressure is 10 20 , we have the 
following interesting table of conclusions regarding the weights 
of atoms and the molecular dimensions of liquefied gases, of water, 
of ice, and of solid metals. 



Substance 


Mass of atom 
or of H S O 
in grammes 


Density 


Number of 
atoms, or 
of molecules 
HjO, in 


Distance in cm. 
from centre 
to centre if 
ranged in cubic 
order with 








cub. cm. 


actual density 


H 


0-45 . 10- 24 


liquid at 17 absolute -090 


200 . 10 21 


1-71 . lO- 8 


O 


7-15 , 


,, ,, freezing point 1-27 


178 


1-78 


H 2 


8-05 


water 1-00 


124 


2-00 


H 2 


8-05 


ice -917 


114 


2-06 


H 2 


8-05 , 


vapour at C. -487 . 10~ 5 


605 . 10 1S 


118-2 


N 


6-29 , 


liquid 1-047 


166 . 10 21 


1-82 


Argon 


17-81 , 


1-42 


79-7 


2-32 


Gold 


88-52 , 


solid 19-32 


218 


66 


Silver 


48-47 


10-53 


217 


1-66 


Copper 


28-43 


8-95 


315 


47 


Iron 


25-15 


7-86 


313 


47 


Zinc 


29-30 


7-15 


245 


60 



81. In the introductory Lecture (p. 14) we considered the 
question " are the vibrations of light perpendicular to, or are they 
" in, the plane of polarization ? defining the plane of polarization 
"as the plane through the incident and reflected rays, for light 
" polarized by reflection." We are now able to answer perpen- 
dicular to the plane of polarization, with great confidence founded 
on two experimental proofs both given by Stokes, and each of 
them alone sufficient, I believe, for the conclusion. 

I. The observed fact that a large proportion of the light of 
the blue sky looked at in any direction perpendicular to the 
direction of the sun, and not too nearly horizontal, is polarized in 
the plane through the sun ; interpreted according to the dynamics 
of Lecture VIII., pp. 88, 89, 90, and of Lecture XVII., 52, 54, 
63. 

II. Observation of the change of the plane of polarization ex- 
perienced by plane polarized light when diffractionally changed in 



STOKES, HOLTZMANN, LORENZ, RAYLEIGH. 323 

direction through a large angle by passage across a Fraunhofer Molar, 
grating; described and interpreted in Part II. of his great paper 
" On the Dynamical Theory of Diffraction*." In a short Appendix, 
pp. 327, 328, added to his Reprint, Stokes notices experiments 
by Holtzmann -f*, published soon after that paper, leading to 
results seemingly at variance with its conclusions, and gives a 
probable explanation of the reason for the discrepance; and he refers 
to later experiments agreeing with his own, by Lorenz of Denmark. 

Thus we find in II. confirmation of the conclusion, first drawn 
by Stokes from I., that the vibrations in plane polarized light 
are perpendicular to the plane of polarization. 

81'. Finally, we have still stronger confirmation of Stokes's 
original conclusion, in fact an irrefragable independent demonstra- 
tion, that the vibrations of light are perpendicular to the plane of 
polarization, by Rayleigh ; founded on a remarkable discovery, 
made independently by himself and Lorenz of Denmark, regarding 
the reflection of waves at a plane interface between two elastic 
solids of different rigidities but equal densities : That, when the 
difference of rigidities is small, and when the vibrations are in 
the plane of incidence, there are two angles of incidence (?r/8 
and 3-7T/8), each of which gives total extinction of the reflected 
light. That is to say ; instead of one " polarizing-angle," tan" 1 /* ; 
there are two " polarizing-angles " 22'5 and 67'5 ; which is utterly 
inconsistent with observation. The old-known fact (proved in 123 
below) that, if densities are equal and rigidities unequal, vibrations 
perpendicular to the plane of incidence give reflected light obeying 
Fresnel's " tangent " law and therefore vanishing when the angle 
of incidence is tan" 1 /*, had rendered very tempting, the false 
supposition that light polarized by reflection at incidence tan" 1 ft 
has its vibrations in the plane of incidence ; but this supposition 
is absolutely disproved by the two angles of extinction of the 
reflected ray which it implies for vibrations in the plane of 
incidence and reflection. 

We may consider it as one of the surest doctrines through the 
whole range of natural philosophy, that plane- polarized light consists 
of vibrations of ether perpendicular to the plane of polarization. 

* Reprint of Mathematical and Physical Papers, Vol. n. pp. 290327. 
t Poggendorff's Annalen, VoL xcix. (1856), p. 446, or Philosophical Magazine, 
Vol. xin. p. 135. 

212 



LECTURE XVIII. 

THURSDAY, October 16, 5 P.M. Written afresh 1902. 

Reflection of Light. 

Molar. I 82. THE subject of this Lecture when originally given was 
" Reflection and Refraction of Light." I have recently found it 
convenient to omit "Refraction" from the title because ( 130 
below), if the reflecting substance is transparent, and if we 
know the laws of propagation of ethereal waves through it, we 
can calculate the amount and quality of the refracted light for 
every quality of incident light, and every angle of incidence, 
when we know the amount and quality of the reflected light. 
When the reflecting substance is opaque there is no " refracted " 
light ; but, co-periodic with the motion which constitutes the 
incident light, there is a vibratory motion in the ether among 
the matter of the reflecting body, diminishing in amplitude 
according to the exponential law, e~ mD , with increasing distance 
D from the interface. When m~ l is equal to 10,000 wave-lengths, 
say half-a-centimetre, the amplitude of the disturbance at one- 
half centimetre inwards from the interface would be e" 1 of the 
amplitude of the entering light, and the intensity would be e~ 2 , 
or 1/7-39, of the intensity of the entering light. The substance 
might or might not, as we please, be called opaque ; but it would 
be so far from being perfectly opaque that both theoretically 
and experimentally we might conveniently deal with the case 
according to the ordinary doctrine of " reflected " and " refracted " 
light. The index of refraction might be definitely measured by 
using very small prisms, not thicker than 1 mm. in the thickest 
part. But, on the other hand, when the opacity is so nearly 
perfect that m~ l is ten wave-lengths, or two or three wave- 
lengths, or less than a wave-length, we have no proper application 
of the ordinary law of refraction, and no modification of it can 
conveniently be used. Omission to perceive this negative truth 
has led some experimenters to waste precious time and work in 
making, and experimenting with, transparent metal prisms of 



TRANSLUCENCE OF METALLIC FILMS. 325 

thickness varying from as nearly nothing as possible at the edge, Molar, 
to something like the thickness of ordinary gold-leaf at distance 
of two or three millimetres from the edge. 

83. A much more proper mode of investigating the pro- 
pagation of ethereal vibrations through metals is that followed 
first, I believe, by Quincke, and afterwards by other able experi- 
menters; in which an exceedingly thin uniform plate of the solid 
is used, and the retardation of phase (found negative for metals 
by Quincke !) of light transmitted through it is measured by the 
well-known interferential method. Some mathematical theorists 
have somewhat marred their work by holding on to " refraction," 
and giving wild sets of real numbers for refractive indices* of 
metals (different for different incidences !), in their treatment of 
light reflected from metals; after MacCullagh and Cauchy had 
pointed out that the main observed results regarding the reflection 
of light from metals can be expressed with some approach to 
accuracy by Fresnel's formulas with fjL = ip + q, where p and q 
are real numbers, and p certainly not zero : q very nearly zero for 
silver, or quite zero for what I propose to call ideal silver ( 150, 
151 below). It was consideration of these circumstances which 
led me to drop " Refraction " from the title of Lecture XVIII. 

84. The theory of propagation of the ethereal motion 
through an opaque solid, due to any disturbance in any part of 
it, including that produced by light incident on its surface, is a 
most important subject for experimental investigation, and for 
work in mathematical dynamics. The theory to be tried for 
( 159) is that of the propagation of vibratory motion through 
ether, when under the influence of molecules of ponderable matter, 
causing the formulas for wave-motion to be modified by making 
the square of the propagational velocity a real negative quantity, 
or a complex. We shall see in 159 below that a new molecular 
doctrine gives a thoroughly satisfactory dynamical theory of 
what I have called ideal silver, a substance which reflects light 
without loss at every incidence and in every polarizational 
azimuth ; its quality in wave-theory being defined by a real 
negative quantity for the square of the propagational velocity 
of imaginary waves through ether within it. Liquid mercury, 

* p is essentially the ratio of the propagational velocities in the two mediums, 
one of which may, with good dynamical reason, be a complex imaginary. See 
144152. 



326 LECTURE XVIII. 

Molar, or quicksilver as it used to be called, may possibly, when tested 
with a very perfectly purified surface, be found to reflect light as 
well as, or nearly as well as, Sir John Conroy succeeded in getting 
a silver surface to do ( 88 below) ; and may therefore fulfil my 
definition of ideal silver : though Rayleigh found only "753 of the 
incident light, to be reflected from fairly clean mercury under air*. 

85. Without any scientific photometry, any one can see that 
the light reflected from gold, steel, copper, brass, tin, zinc, however 
well polished, is very much less than from silver, and hence very 
. much less than the whole incident light. Though many excellent 
investigations have been made by many able experimenters on 
the polarizational analysis of light reflected from all these metals, 
few of them have given results as to the proportion of the whole 
reflected, to the whole incident, light. It seems indeed that, 
besides Rayleigh, only two observers, Potterf and Conroy J, have 
directly compared the whole light reflected from metals with the 
whole incident light (see 88 below). Conroy, using his most 
perfect silver mirror, and angle of incidence 30, found the propor- 
tion of reflected light to incident light to be 97'3 percent., for light 
vibrating perpendicular to the plane of incidence, and 99'9 per 
cent, for light vibrating in the plane of incidence ; and therefore 
98'6 per cent, for common un polarized light. With speculum-metal 
and steel and tin mirrors, for 30 of incidence, and unpolarized 
light, he found the reflected light to be; speculum-metal 66'9 
per cent., steel 54'9 per cent., and tin 44'4 per cent.|| From these 
measurements, as well as from ordinary non-scientific observation, 
we see that there is essentially much light lost in reflection from 
other metals than silver; and, as the light does not travel through 
the metal, it is quite certain that its energy must be converted 
into heat in an exceedingly thin surface-region of the metal 
(certainly not more than two or three wave-lengths). Hence the 
square of the imaginary velocity of the imaginary light-wave in 
other metals than silver cannot be a real negative quantity; 

* Phil. Mag. 1892 ; Collected Papers, Vol. iv. 

t Edinburgh Journal of Science, Vol. in. 1830, pp. 278288. 

t Proc. It. S. Vol. xxvin. Jan. 1879, pp. 242250 ; Vol. xxxi. Mar. 1881, 
pp. 48G 500; Vol. xxxv. Feb. 1883, pp. 2641; Vol. xxxvi. Jan. 1884, pp. 187 
198; Vol. xxxvn. May, 1884, pp. 3642. 

I have found no other recorded case of greater reflectivity of vibrations in 
than of vibrations perpendicular to the plane of incidence. 

|| Proc. E. S. Feb. 1883, Vol. xxxv. pp. 31, 32. 



MOLECULARLY SMOOTH; NEAREST TO PERFECT POLISH. 327 

because, as we readily see from the mathematical treatment, this Molar, 
would imply no conversion of light-energy into heat, and would 
therefore ( 150, 157 below) imply total reflection at every angle 
of incidence. Dynamical theory is suggested in 159, to explain 
conversion of incident light into heat, among the molecules 
within two or three wave-lengths of the boundary of the metallic 
mirror. This is only an outlying part of the whole field of in- 
vestigation required to explain all kinds of opacity in all kinds 
of matter, solid, liquid, and gaseous. 

86. Let us think now of the reflection of light from perfectly 
polished surfaces. If the substance is infinitely fine-grained the 
polish may be practically perfect ; so that no light is reflected 
except according to the law of equal angles of incidence and 
reflection. But in reality the molecular structure of solids gives 
a surface which is essentially not infinitely fine-grained : and the 
nearest approach to perfect polish produced by art, or found in 
nature, on the surfaces of liquids and on crystalline or fractured 
surfaces of solids, is illustrated by a gravel-covered road made as 
smooth as a steam-roller can make it. Optically, the polish would 
be little less than perfect if the distances between nearest neigh- 
bours in the molecular structure are very small in comparison with 
the wave-length of the incident light. In 80 we estimated the 
distances between nearest molecular neighbours in ordinary solids 
and liquids at from 1-5 . 10~ 8 to 2 . 10~ 8 cm., which is from 1/4000 
to 1/3000 of the mean wave-length of visible light (6 . 10~ 5 cm.). 
The best possible polish is therefore certainly almost quite practi- 
cally perfect in respect to the reflection of light. But if the work 
of the steam-roller is anything less than as perfect as it can pos- 
sibly be, there are irregular hollows (bowls or craters) of breadths 
extending, say, to as much as 1/200 of the wave-length of mean 
visible light; and the polish would probably not be optically perfect. 
The want of perfectness would be shown by a very faint blue light, 
scattered in all directions from a polished mirror illuminated only 
by a single lamp or by the sun. The best way to look for this 
blue light would be to admit sunlight into an otherwise dark room 
through a round hole in a window-shutter or a metal screen, as 
thoroughly blackened as possible on the side next the room ; and 
place the mirror to be tested in the centre of the beam of light at 
a convenient distance from the hole. If the polish is optically 
perfect, no light is seen from any part of the mirror by an eye 



328 LECTURE XVIII. 

Molar, placed anywhere not in the course of the properly reflected light 
from sun or sky. That is to say, the whole room being dark, and 
the screen around the hole perfectly black, the whole mirror would 
seem perfectly black when looked at by an eye so placed as not to 
see an image (of the hole, and therefore) of any part of the sky or 
sun. A condition of polish very nearly, but not quite, optically 
perfect, would be shown by a faint violet-blue light from the 
surface of the mirror instead of absolute blackness. The tint of 
this light would be the true Rayleigh X~ 4 blue, if the want of per- 
fectness of the polish is due to craters small in comparison with 
the wave-length (even though large in comparison with distances 
between nearest molecular neighbours). When this condition is 
fulfilled, the blue light due to want of perfectness in the polish, 
seen on the surface if viewed in any direction perpendicular to the 
direction of the incident beam, would be found completely polar- 
ized in the plane of these two directions. Everything in 86 is 
applicable to reflection from any kind of surface, whether the 
reflecting body be solid or liquid, or metallic, or opaque with any 
kind of opacity, or transparent. Experimental examination of the 
polish of natural faces of crystals will be interesting. 

87. Valuable photometric experiments with reference to 
reflection of light by metals have been made by Bouguer, Biot, 
Brewster, Potter, Jamin, Quincke, De Senarmont, De la Provostaye 
and Desains, and Conroy*. But much more is to be desired, not 
only as to direct reflection from metals, but as to reflection at 
all angles of incidence from metals and other opaque and trans- 
parent solids and liquids. In each case the intensity of the 
whole reflected light should be compared with the intensity of 
the whole incident light ; in the first place without any artificial 
polarization of the incident, and without polarizational analysis 
of the reflected, light. It is greatly to be desired that thorough 
investigation of this kind should be made. It would be quite an 
easy kind of work-f- because roughly approximate photometry is 

* Bouguer, Traite d'Optique, 1760, pp. 27, 131: Biot, Ann. de Chimie, 1815, 

, VoJ. xciv. p. 209: Brewster, A Treatise on new Philosophical Instruments, Edin. 

1^13, p. 347 ; Phil. Tram. 1830, p. 69 : Potter, Edin. Jour, of Science, 1830 : De la 

; Trovostaye and Desains, Ann. de Chim. et de Phys. [3], 1849, Vol. xxvn. p. 109, and 

1850, Vol. xxx. pp. 159, 276: Conroy, Proc. R. S. ; see 84 above. See Mascart, 

Traite d'Optique, Vol. n. 1891, pp. 441459. 

t Bouguer, Traite d'Optique, 1760, pp. 27, 131 : Arago, (Euvres Completes, Vol.*. 
pp. 150, 185, 216, 468. See Mascart, Traite d'Optique, Vol. n. 1891, pp. 441501. 



REFLECTIVITIES; FRESNEL, POTTER, JAMIN, AND CONROY. 329 



always easy, except when rendered impossible by difference of Molar, 
colour in the lights to be compared. 

Absolute determinations of reflected light per unit of incident 
light cannot be made with great accuracy because of the inherent 
difficulty, or practical impossibility, of very accurate photometric 
observations, even when, as is largely the case for reflected lights, 
there is no perceptible difference of tint between the lights to be 
compared. 

88. The accompanying diagram, fig. 1, shows, for angles of 
incidence from 10 to 80, the reflectivities* of silver, speculum- 



Spe 








20 



90 



40 50 
Fig. 1. 



90" 



* I propose the word "reflectivity " to designate the ratio of the whole reflected 
light to the whole incident light, whether the incident light is unpolarized as in 
fig. 1; or plane polarized, either in or perpendicular to the plane of incidence, 
as in figs. 4 7 ; and whether it is ordinary white light, or homogeneous light, or 
light of any mixed tint. 



330 LECTURE XVIII. 

Molar, metal, steel, and tin, according to the observations of Potter, 
Conroy, and Jamin ; and for all incidences from '0 to 90 the 
reflectivities of diamond, flint-glass, and water, calculated from 
Fresnel's admirable formulas. These are, for almost all trans- 
parent bodies even of so high refractivity as diamond, ( 96, 100 
below), probably very much nearer the truth than any photo- 
metric observations hitherto made, or possible to human eyes. 
The ordinates represent the reflected light in percentage of the 
incident light, at the angles of incidence represented by the 
abscissas. The reflectivities thus given for the three transparent 
bodies are the means of the reflectivities given in figs. 4, 5, 6 of 102 
below for light polarized in, and perpendicular to, the plane of 
incidence and reflection. Jamin's results* for steel are given by 
himself as the means of the reflectivities for light polarized in, 
and perpendicular to, the plane of incidence and reflection ; each 
determined photometrically by comparison with reflectivities of 
glass calculated from Fresnel's formulas. The six curves for metals, 
of this diagram (fig. 1), show I believe all the reflectivities that 
have hitherto been determined by observation ; except those of 
Jaminf for normal incidence of homogeneous lights from red to 
violet on metals, and Rayleigh'sJ for nearly normal incidence of 
white light on mercury and glass. All the curves meet, or if 
ideally produced meet, in the right-hand top corner, showing total 
reflection at grazing incidence. 

89. The accompanying sketches (figs. 2 a, 26) represent 
what (on substantially the principle of Bouguer, Potter, and 
Conroy) seems to me the best and simplest plan for making 
photometric determinations of reflectivity. The centres of lamps, 
mirror, and screen, are. all in one horizontal plane, which is taken 
as the plane of the drawings. The screens and reflecting face of 
the mirror are vertical planes. R is the reflecting surface, shown 
as one of the faces of an acute-angled prism. L, L' are two as 
nearly as may be equal and similar lamps (the smaller the 
horizontal dimensions the better , provided the light shed on the 

* Mascart, Traite d'Optiqiie, Vol. n. pp. 534, 536. t Ibid. p. 544. 

J Scientific Papers, Vols. n. and iv. ; Proc. R. S. 1886, Phil. Mag. 1892. 

The best non-electric light I know for the purpose is the ordinary flat-wicked 
paraffin-lamp, with a screen having a narrow vertical slot placed close to the lamp- 
glass, with its medial line in the middle plane of the flat flame. The object of the 
screen is to cut off light reflected from the glass funnel, without intercepting any 
of the light of the flame itself; the width of the slot should therefore be very 



EXPERIMENTAL METHOD FOR REFLECTIVITIES. 



331 



mirror is sufficient). L is too far from R to allow its true position Molar, 
to be shown on the diagram, but its direction as seen from R is 
indicated by a broken straight line. W is an opaque screen coated 




Fig. 2fe. 



Fig. 2 a. 

slightly greater than the thickness of the flame. This arrangement gives a much 
finer and steadier line of light than any unscreened candle, and vastly more light; 
and it gives more light than comes through a slot of the same width from a 
round-wicked paraffin -lamp. An electric lamp of the original Edison "hair-pin" 
pattern, with a slotted screen to cut off reflected light from the glass, would be 
more convenient than the paraffin-lamp, and give more light with finer shadows. 



332 LECTURE XVIII. 

Molar, with white paper on the side next E, the eye of the observer. 
Fig. 2 a and fig. 2 b show respectively the positions for very 
nearly direct incidence, arid for incidence about 60. Each 
drawing shows, on a scale of about one-tenth or one-twentieth, 
approximately the dimensions convenient for the case in which 
the mirror is a surface of flint-glass of refractive index T714; 
and, as according to Fresnel's theory, represented by the mean of 
the ordinates of curves 1 and 2 of fig. 6 ( 102 below), having re- 
flectivities 1/14 and 1/4 at incidences and GO . The eye at E 
sees a narrow portion of the white screen next the right-hand edge 
illuminated only by I, the image of L in the mirror; and all of the 
screen on the left of that portion, illuminated by the distant lamp 
L'. The distant lamp is moved nearer or farther till the light is 
judged equal on the two sides of the border line between the 
illumination due to I, the image of L, and the direct illumination 
due to L'. By shifting L' slightly to the right, or slightly to the 
left, we may arrange to have either a very narrow dark space, or 
a space of double brightness, between the two illuminations ; and 
thus the eye is assisted in judging as to the perfect equality of 
the two. When, as in fig. 26, the angle of incidence exceeds 
45 a dark screen DD is needed to prevent the light of L from 
shining directly on the white screen, W. The method, with the 
details I have indicated, is thoroughly convenient for reflecting 
solids, whether transparent or metallic or of other qualities of 
opacity ; provided the mirror can be made of not less than two 
or three centimetres breadth. But, as in the case of diamond, 
when only a very small mirror is available, modification of the 
method to allow direct vision of the lights to be compared (with- 
out projection on a white screen) would be preferable or necessary. 
For liquids, of course, modification would be necessary to suit it 
for reflection in a vertical plane. 

90. As remarked at the end of 87, absolute determinations 
of reflectivities cannot be made with great accuracy, because of the 
imperfectness of perceptivity of the eye in respect to relative 
strengths of light, even when the tints are exactly the same. On 
the other hand, a very high degree of accuracy is readily attainable 
by the following method*, when the problem is to compare, for 
any or every angle of incidence, the reflected lights due to the 

* Given originally by MacCullagh. See his Collected Works, p. 239; also 
Stokes, Collected Works, Vol. m. p. 199. 



POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 333 

incidence of equal quantities of light vibrating in, and vibrating Molar, 
perpendicular to, the plane of incidence. 

Use two Nicol's prisms, which, for brevity, I shall call N t and 
N%, in the course of the incident and reflected light respectively. 
Use also a Fresnel's rhomb (F) between the reflecting surface 
and N t . Set N 3 , and keep it permanently set, with its two 
principal planes at 45 to the plane of reflection, but with facilities 
for turning from any one to any other of the eight positions thus 
defined, to secure any needful accuracy of adjustment. For brevity, 
I shall call the zeros of .AT, and F, positions when their principal 
planes are at angles of 45 to the plane of incidence and 
reflection. In the course of an observation JVj and F are to be 
turned through varying angles, n and f, from their zeros, till 
perfect extinction of light coming through N, is obtained. 

91. To begin an observation, with any chosen angle of 
incidence of the light on the reflecting surface, turn N t to a 
position giving as nearly as possible complete extinction of light 
emerging from N,. Improve the extinction, if you can, by turning 
F in either direction, and get the best extinction possible by 
alternate turnings of N r and F. Absolutely complete extinction 
is thus obtained at one point of the field if homogeneous light 
is used, and if the Nicols and rhomb are theoretically perfect 
instruments. The results of the completed observation are the 
two angles (n, f) through which N l and F must be turned from 
their zero positions to give perfect extinction by N t . From f 
thus found we calculate by two simple formulas, (7) of 93, 
the ratios of the vibrational amplitudes of the two constituents 
defined below, (< g\ of the reflected vibrations in the plane of 
reflection, to the vibrational amplitude (C) of the reflected vibra- 
tions perpendicular to that plane. The definite constituents here 
referred to are () vibrations in the same phase as (7; and (g) 
vibrations in phase advanced by a quarter-period relatively to (7. 
It is clear that if g = 0, complete extinction would be had without 
turning F from its zero position, and would be found by the 
same adjustment of N t as if there were no Fresnel's rhomb in the 
train. (C corresponds to the C' of 117, 123, below.) 

92. Figs. 3a, 36, are diagrams in planes respectively perpen- 
dicular to the reflected, and to the incident, light. Let be a point 
in the course of the light between the reflecting surface and the 



334 



LECTURE XVIII. 



Molar. Fresnel's rhomb. Let OG be the plane of reflection of the light from 
. the reflecting surface, and let OZ be perpendicular to that plane. 

O 




Let 0^ be the vibrational plane* of the second Nicol; and OP 
its "plane of polarization " (being perpendicular to ON 2 ). Let OF 
be the plane through the entering ray, and perpendicular to the 
facial intersections f of the Fresnel's rhomb ; shown in the diagram 
as turned through an angle f= N 2 OF from the zero position 
ON 2 . Let OK be perpendicular to OF. In respect to signs we 
see by 158 V (1) that in fig. 3a, for reflected, and 36 for incident, 
vibrations, OG is positive. 

Considering the light coming from the reflecting surface and 
incident on the Fresnel, let C sin wt be the component along OZ, 
and sin &> g cos wt be the component along OG, of the dis- 
placement at time t of a particle of ether of which is the 
equilibrium position. </C and gfC are two functions of the angle 
of incidence to be determined by the observation now described. 
By proper resolutions and additions for vibrational components 
in the principal planes of the Fresnel, we find as follows : 

fir 
A 






and 



p cos fa - /) - Csm (^ -/)J along 

[fir \ fir \~} 

^in(j-/) + Cco S (^-/)J ,, 

cos cot g cos f -j /] along OF \ 



OK 



...(1), 



cos tot g sin ( - 



OK 



* I use this expression for brevity to denote the plane of the vibrations of light 
transmitted through a Nicol. 

t An expression used to denote the intersections of the traversed faces and the 
reflecting surfaces. See 158" below. 



POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 335 

By the two total internal reflections at the oblique faces of the Molar. 
Fresnel, vibrational components in the plane OF are advanced a 
quarter-period relatively to the vibrational components perpen- 
dicular to OF. Hence to find the vibrational components at time 
t at a point in the course of the light emerging from the Fresnel, 
we must, in the components along OF o( (1) and (2), change the 
sin a)t into cos tat, and change the cos tat into sin tat ; and leave 
unchanged the components along OK. Thus we find for the 
vibrational components at the chosen point in the course of the 
light from the Fresnel towards the second Nicol, as follows : 



sin tat g cos C^ A along OF 

I /g\ 

sin tat I c^sin fc-A+C cos (^ -/) 1 OJST 
and 

cosw* j^cos ^ -/) - (7 sin (j -/) | alon g #** 



* 08'n(j-/) 01T 



...(4). 



93. For extinction by the second Nicol, the sum of all the 
vibrational components parallel to ON 2 of the light reaching it 
must be null ; and therefore, by the proper resolutions and 
additions, and by equating to zero separately the coefficients 
of sin tot and cos tat thus found, we have 

- 1^ sin ^ -/) + C cos f ^ -/) 1 sin/+# cos ^ - fj cos/= 

(5), 

T /TT \ fir \~\ fir \ 

cos ( -T f\ (7 sin ( - /) COS/+ g sin f j /J sin/ = 

(6). 

Solving these equations for and g, we find 

2sin2/ 



94. Go back now to the light emerging from the first 
Nicol and incident on the reflecting surface. Let / be its 



336 



LECTURE XVIII. 



Molar, vibrational amplitude. The vibrational amplitudes of its com- 
ponents in, and perpendicular to, the plane of incidence are 



T (""" 

1 cos -r n 



and / sin f n 

when the Nicol is turned from its zero position, OQ, through the 
angle n, as indicated in fig. 36 ; ON-,, being the vibrational plane of 
the first Nicol. 




94'. Let 8 be the ratio of reflected to incident vibrational 
amplitude for vibrational component perpendicular to the plane 
OG. Considering next the component of the incident light having 
vibrations in the plane OG ; let T and E be the ratios of the vibra- 
tional amplitudes of two particular constituents of its reflected 
light both vibrating in the plane OG, to the vibrational amplitude 
of the incident component ; these two constituents being respec- 
tively in the same phase as the component of the reflected light 
vibrating perpendicular to OG, and in phase behind it by a quarter- 
period*. We have 



From these and (7) we have 



+ cos 4/ 

A tan - - n 



# - 2 sin 2/ 



* When the reflecting body is glass, or other transparent isotropic solid or 
liquid, Fresnel's prophecy (we cannot call it physical or dynamical theory) declares 
_ sin (i - t i) _ tan (i - ,i) _ 

sin (i + ,i) ' tan (i + t i ) ' 

The notation in the text is partially borrowed from Rayleigh (Scientific Papers, 
Vol. in. pp. 496512) who used S, and T, to denote respectively the "sine-formula," 
and the "tangent-formula," of Fresnel. What I have denoted in the text by E is, 
for all transparent solids and liquids, certainly very small; and though generally 



POLARIZATION A L ANALYSIS OF REFLECTED LIGHT. 337 

Thus our experimental result (/, ) gives the two constituents Molar. 
(T, E) of the reflected vibrations in the plane of reflection, when 
the single constituent (5) of the reflected vibrations perpendicular 
to the plane of reflection, is known. 

95. Going back to 90, note that there are four independent 
variables to be dealt with : the angle of incidence, the orientations 
of the two Nicol's prisms, and the orientation of the Fresnel's 
rhomb. Any two of these four variables may be definitely chosen 
for variation while the other two are. kept constant; to procure, 
when homogeneous light is used, extinction of the light which 
enters the eye from the centre of the field ; that is to say, to 
produce perfect blackness at the central point of the field. Theo- 
retically there is, in general, just one point of the field (one point 
of absolute blackness) where the extinction is perfect ; and always 
before the desired adjustment is perfectly reached a black spot is 
conceivably to be seen, but not on the centre of the field. By 
changing any one of the four independent variables the black spot 
is caused to move; and generally two of them must be varied 
to cause it to move towards the centre of the field for the desired 
adjustment. What in reality is generally perceived is, I believe, 
not a black spot, but a black band ; and this is caused to travel 
till it passes through the middle of the field when the nearest 
attainable approach to the desired adjustment is attained. When 
faint or moderate light, such as the light of a white sky, is used, 
the whole field may seem absolutely dark, and may continue so 
while any one of the four variable angles is altered by half a 
degree or more. For more minutely accurate measurements, more 
intense light must be used ; a brilliant flame ; or electric arc- 
light ; or lime-light; or, best of all, an unclouded sun as in 
Rayleigh's very searching investigation of light reflected from 
water at nearly the polarizing angle, of which the result is given 
in 105 below. 

95'. An easy way to see that just two independent variables are 
needed to obtain the desired extinction, is to confine our attention 
to the centre of the field, and imagine the light reaching the eye 

believed to be perceptible for substances of high refrangibility such as diamond, 
Rayleigh has questioned its existence for any of them, and suggested that its non- 
nullity may be due to extraneous matter on the reflecting surface. I believe, however, 
that Airy, and Brewster, and Stokes who called it the adamantine property, and 
Fresnel himself though it was outside his theory, were right in believing it real. 
See 158, below. 

T.L. 22 



338 LECTURE XVIII. 

Molar, from it when the extinction is not perfect, to be polarizationally 
resolved into two components having their vibrational lines 
perpendicular to one another. The desired extinction requires the 
annulment of each of these two components, and nothing else. 
Proper change of the two chosen independent variables deter- 
minately secures these two annulments when what is commonly 
called homogeneous light, that is, light of which all the vibrations 
are of the same period, is used. 

95". In the detailed plan of 90 94 the two independent 
variables chosen are the orientation of the first Nicol, and the 
orientation of the Fresnel's rhomb. This is thoroughly convenient if 
N 2 is mounted on a proper mechanism to give it freedom to move 
in a plane perpendicular to its axis, and to keep its orientation 
round this axis constant. The Fresnel should be mounted so as 
to be free to move round an axis fixed in the direction of the light 
entering it: and it should carry a short tube round the light 
emerging from it into which N 2 fits easily. Thus, when the 
Fresnel is turned round the line of the light reflected from the 
mirror, it carries N* round in a circle (as it were with a hollow 
crank-pin), so that it is always in the proper position for carrying 
the emergent light to the eye of the observer. 

Two other choices of independent variables, each, I believe, 
as well-conditioned and as convenient as that of 90 94, but 
simpler in not wanting the special mechanism for carrying N 2 , 
are described in 98 below. 

96. Nothing in 90 95 involves any hypothesis : and we 
have, in them, an observational method for fully, without any 
photometry, determining T/S and EJ8\ which are, for incident 
vibrations at 45 to the plane of incidence, the intensities of the 
two constituents of the reflected light vibrating in the plane of 
incidence, in terms of the intensity of the component of the re- 
flected light vibrating perpendicularly to that plane. Fully carried 
through, it would give interesting and important information, for 
transparent liquids and solids, and for metals and other opaque 
solids, through the whole range of incidence from to 90. It can 
give extremely accurate values of T/S for transparent liquids and 
solids; and it will be interesting to find how nearly they agree 

with the formula ^/*_'*y which Fresnel's "tangent-law" and 
" sine-law " imply. 



POLARIZATIUNAL ANALYSIS OF REFLECTED LIGHT. 339 

90'. Fresnel himself used it* for reflection from water Molar, 
and from glass with incidences from 24 to 89 (not using the 

T 
Fresnel's rhomb), and found values for tan" 1 ~ differing from 

tan" 1 - r^ by sometimes more than 1; but it has been 

COS \l ,1 ) 

supposed that these differences may be explained by the im- 
perfection of his apparatus, and by the use of white lightf. 
Similar investigation was continued by BrewsterJ, on several 
species of glasses and on diamond. With, for example, a glass of 

T 

refractive index 1'4826, his observed results for tan" 1 ^ differed 

o 

from tan- 1 - 4^ by 1 4'; which he considered might be 

COS \l ,\) 

within the limits of his observational errors. For diamond, he 
found greater deviations which seemed systematic, and not errors 
of observation. With a Fresnel's rhomb used according to the 
method of 90 94 he might probably have found the definite 
correction on Fresnel's formula, required to represent the polari- 
zational analysis of reflection from diamond. Can it be that 
both Fresuel and JBrewster underestimated the accuracy of their 
own experiments, and that even for water and glasses, devia- 
tions which they found from Fresnel's - '-{ may have been 

cos(t-,t) 

real, and not errors of observation ? The subject urgently demands 
full investigation according to the method of 90 98, with all 
the accuracy attainable by instruments of precision now available. 

97. The reader may find it interesting to follow the formulas 
of 94 through the whole range of incidences from to 90. 
For the present, consider only the case of the angle of incidence 
which makes T=0. This, being the incidence which, when the 
.incident light is polarized in any plane oblique to the plane of 
incidence, gives 90 difference of phase for the components of the 
reflected light vibrating in that plane and perpendicular to it, has 
been called by Cauchy, and I believe by all following writers, the 
principal incidence. We shall see presently ( 97", 99, 105) that, 
for every transparent substance, observation and dynamics show one 
incidence, or an odd number of incidences, fulfilling this condition. 

* Fresnel, (Euvret, Vol. i. p. 646. 

t Mascart, Trait* tPOptique, Vol. H. p. 466. 

t Brewster, Phil. Tram. 1830. See also Mascart, Vol. u. p. 406. 

222 



340 LECTURE XVIII. 

Molar. By observation and dynamics we learn (81 above), that it is 
fulfilled not by the vanishing of S, but by the vanishing of T. 
To make T = we have by (9) for the case of principal incidence, 
1 + cos 4/= 0, and therefore /= 45. This, by (9), makes (if we 
takey= + 45) for Principal Incidence, 

E/S = tan (n - Tr/4) = A; (10). 

97'. The k here introduced is Jamin's notation, adopted 
also by Rayleigh. It is the ratio of the vibrational amplitude of 
reflected vibrations in the plane of incidence to the vibrational 
amplitude perpendicular to it, when the incident light is polarized 
in a plane inclined at 45 to the plane of incidence, and when the 
angle of incidence is such as to make the phases of those two com- 
ponents of the reflected light differ by 90. k is positive or 
negative according as the phase of the reflected vibration in the 
plane of incidence lags or leads by 90 relatively to the component 
perpendicular to it. It is positive when (as in every well assured 
case* whether of transparent or of metallic mirrors) the obser- 
vation makes n > 45 ; it would be negative if n < 45. 

97". The angle n 7r/4 found by our observation of 97 is 
called the "Principal Azimuth." See 158 xvii below. It has been 
the usage of good writers regarding the polarization of light, par- 
ticularly in relation to reflection and refraction, to give the name 
"azimuth^" to the angle between two planes through the direction 
of a ray of light ; for instance, the angle between the plane of in- 
cidence and the plane of vibration of rectilineally polarized light. 
A "Principal Azimuth," for reflection at any polished surface, I 
define as the angle between the vibrational plane of polarized light 
incident at Principal Incidence, and the plane of the incidence, 
to make the reflected light circularly polarized. There is one, and 
only one, Principal Incidence for every known mirror : except 
internal reflection in diamond and other substances whose refrac- 
tive indices exceed 2'414; these have three Principal Incidences 
( 158"' below). The number is essentially odd: on this is founded 
the theory of the polarization of light by reflection. 

* See Jamin, Cours de Physique, Vol. n. pp. 694, 695 ; also below, 105, 154, 
158 V , 159"', 179, 192. 

t This, when understood, is very convenient ; though it is not strictly correct. 
Azimuth in astronomy is essentially an angle in a horizontal plane, or an angle 
between two vertical planes. A reader at all conversant with astronomy would 
naturally think this is meant when a writer on optics uses the expression 
"Principal Azimuth," in writing of reflection from a horizontal mirror. 



PRINCIPAL INCIDENCE AND PRINCIPAL AZIMUTH. 341 

98. The vibrational plane of the incident light is inclined Molar, 
to the plane of incidence at an angle of n 45 ; which, for the 
Principal Incidence, is such as to render the two components 
of the reflected light equal; and therefore to make the light 
circularly polarized. However a Fresnel's rhomb is turned, circu- 
larly polarized light entering it, leaves it plane polarized. In 
the observation, with the details of 90 94, it is turned so 
that the vibrational plane of the light emerging from it is per- 
pendicular to the fixed vibrational plane of N z . Hence it occurs 
to us to think that a useful modification of those details might 
be; to fix the Fresnel's rhomb with its principal planes at 45 
to the plane of reflection, and to mount N 2 so as to be free 
to turn round the line of light leaving the Fresnel's rhomb. 
Alternate turnings of N t and N^ give the desired extinction for 
any angle of incidence. Take /= in (1), (2), (3), (4); which 
makes G0^=45 in fig. 3 a. Let n^ be the angle (clockwise in 
the altered fig. 3 a) from OF to ON^ and put n = ITT + a in fig. 3 6. 
Eliminating and g from (3), (4), by (8) and proceeding as in 
93, we find 

iy,S= tan a cos 2w,; E/8 = - tan a sin 2w 2 ; E/T=- tan2H 2 ...(10'). 

If a is positive observation makes n^ negative when taken acute. 
For principal incidence it is 45, and E/S = tan a. The phasal 
lag of (E, T) behind (S) is - 2nj. Negative is anti-clockwise in 
fig. 3 a. See 158"". 

99. The following table shows, for six different metals, 
determinations of principal incidences and principal azimuths 
which have been made by Jamin and Conroy, experimenting 
on light from different parts of the solar spectrum. This table 
expresses, I believe, practically almost all that is known from 
observations hitherto made as to the polarizational analysis of 
homogeneous light reflected from metals. The differences between 
the two observers for silver are probably real, and dependent on 
differences of condition of the mirror-surfaces at the times of 
the experiments, as modified by polishing and by lapses of time. 
It will be seen that for each colour Jamin's k? is intermediate 
between Conroy 's two values for the same plate, after polishing 
with rouge and with putty powder respectively. On the other 
hand, each of Conroy's Principal Incidences for silver is greater 



342 



LECTURE XVIII. 



* 



5 
ll 

tl 



i^- 't* 



CM M 

23 



.g 



00 

1^ O O 

CD W 



CM O I- 



s 



<i b 

>C CN 



1 



CD 05 W -t 



!^ -01.- g 

rTn ii ifii H 



POLARIZATIONAL ANALYSIS OF REFLECTED LIGHT. 343 

than Jamin's; and by greater differences for the yellow and blue Molar, 
light than for the red. 

Experimental determinations of T/S and E/S, (9) or (KX), 
through the whole range of incidence below and above the 
Principal Incidence are still wanting. 

100. As for transparent solids and liquids, we may consider 
it certain that Fresnel's laws, giving 

sin(t-,t) T tan(t-,t) ~ n 

-^ = sln7iT7r ~ T= tan (; + ,')' E = Q <">' 

are very approximately true through the whole range of incidence 
from to 90; but, as said in 95, it is still much to be desired 
that experimental determinations of T/S should be made through 
the whole range ; in order either to prove that it differs much less 

from f-. '-* than found experimentally by Fresnel himself and 

COS (I t l) 

Brewster; or, if it differs discoverably from this formula, to deter- 
mine the differences. It is certain, however, that at the Principal 
Incidence the agreement with Fresnel's formula (implying E= in 
the notation of 94') is exceedingly close ; but the very small 
deviations from it found experimentally by Jamin and Rayleigh 
and represented by the values of k shown in the table of 105 
below, are probably real. An exceedingly minute scrutiny as to 
the agreement of the Principal Incidence with tan~>, Brewster's 
estimate of it; a scrutiny such as Rayleigh made relatively to 
the approach to nullity of k for purified water surfaces ; is still 
wanted; and, so far as I know, has not hitherto been attempted 
for water or any transparent body. See 180, 182 below. 

Hitherto, except in 81, 84, 86, we have dealt exclusively 
with what may be called the natural history of the subject, and 
have taken no notice of the dynamical theory ; to the considera- 
tion of which we now proceed. 

101. Green's doctrine* of incompressible elastic solid with 
equal rigidity, but unequal density, on the two sides of an interface, 
to account for the reflexion and refraction of light, brings out for 
vibrations perpendicular to the plane of incidence ( 123 below) 
exactly the sine-law which Fresnel gave for light polarized in the 
plane of incidence. On the other hand, for vibrations in the 

Camb. Phil. Soc., Dec. 1837 ; Green's Collected Papers, pp. 246, 258, 267, 268. 



344 LECTURE XVI11. 

Molar, plane of incidence it gives a formula ( 104, 105, 146 below) 
which, only when the refractive index differs infinitely little from 
unity, agrees with the tangent-law given by Fresnel for light 
polarized perpendicular to the plane of incidence; but differing 
enormously from Fresnel, and from the results of observation, in 
all cases in which the refractive index differs sufficiently from 
unity to have become subject of observation or measurement. 

102. The accompanying diagrams, figs. 4, 5, 6, illustrate, 
each by a single curve (Curve 1), the perfect agreement between 
Green and Fresnel for the law of reflection at different incidences 
when the vibrations are perpendicular to the plane of incidence ; 
and by two other curves the large disagreement when the vibra- 
tions are in the plane of incidence. 

Curve 1 in each diagram shows for vibrations perpendicular to 
the plane of incidence the ratio of the reflected to the incident 
light according to Fresnel's sine-law 

pan (*-,) 
[sin (+,)_ 

dynamically demonstrated by Green on the hypothesis of equal 
rigidities and unequal densities of the two mediums. 

Curve 2 shows, for vibrations in the plane of incidence, the 
ratio of the reflected to the incident light according to Fresnel's 
tangent-law, 

"tan (i- ,iy** 



Curve 3 shows, for vibrations in the plane of incidence, the 
ratio of reflected to incident activity (rate of doing work) per 
unit area of wave-plane, rigorously demonstrated by Green ( 146 
below) for plane waves incident on a plane interface between 
elastic solids of different densities but the same rigidity; on the 
supposition that each solid is absolutely incompressible, and that 
the two are in slipless contact at the interface. 

In each diagram abscissas from to 90 represent " angles of 
incidence," that is to say, angles between wave-normals and the 
line-normal to the interface, or angles between the wave-planes 
and the interface. 

103. Curves 2 and 3 of fig. 4 show for water a seemingly fair 
agreement between Fresnel and Green for vibrations in the plane 



FRESXEL, P.REEN, AND TUt'Tll. 



345 



of reflection. But the scale of the diagram is too small to show Molar, 
important differences for incidences less than 60 or 65, especially 
in the neighbourhood of the polarizing angle, 53'l : this want is 
remedied by the larger scale diagram fig. 7 showing Curves 2 and 
3 of fig. 4; on a scale of ordinates 48*5 times as large, in which, 
for vibrations in the plane of incidence, the unit for intensity 
of light is the reflected light at zero angle of incidence, instead 







































































































1 


































1 


































/i 


































ffl 


































1 


































^ 


































// 


































/I 


































^ 1 
































1 


3 
































1 


/ 
































1 


// 






























/ 




l\ 






























/ 


l\ 




























, 


7 




II 


























c^ 


X 




fl? 


*fa' 




F 1 


a 

~~i 


i ; 

CT~ 


2 


. . 

r 


3 

91 




5^" 

t i 



i 
4 

, ^ 


- 

a"" 

'at. 


^r^ 

ti ' 
5 

r. 





~6 

i*aa 


( 
=^ 

0" 

1) 


7 


^ 

o 3 " 


\ 
~~8 


? 9 



of the incident light at incidence i as in the other diagrams. 
Curve 2 in figs. 4 and 7 shows, for water, the absolute extinction 
at angle of incidence tan~y given by Fresnel's formula. Curve 3 
(Green's formula) shows, for a slightly smaller angle of incidence 
(50 '0 instead of 53'l), a minimum intensity equal to '295 of 
th.u of directly reflected light; that is to say Green's formula 



346 



LECTURE XVTTI. 



Molar, makes the directly reflected light from water only 3 times 
as strong as the light reflected at the angle which gives least 
of it. 

104. To test whether Fresnel or Green is more nearly right, 
take a black japanned tray with water poured into it enough to 
cover its bottom, and look through a Nicol's prism at the image 




10 20 30 40 50 80 70 80 90 

Fig. 5. Flint Glass. (M=1'714) 

of a candle in the water-surface. You will readily find in half-a- 
minute's trial a proper inclination of the light and orientation of 
the Nicol to give what seems to you extinction of the light. To 
test the approach to completeness .of the extinction let an assistant 
raise and lower alternately a piece of black cloth between the candle 
and the water surface, taking care that it is lowered sufficiently 
to eclipse the image of the candle when it is not extinguished by 



FRKSNEL, OREEN, AND TRUTH. 



847 



the Nicol. By holding the Nicol very steadily in your hand, and Molar, 
turning to give the best extinction you can produce by it, you 
will find no difference in what you see whether the screen is down 
or up, which proves a very close approach to perfect extinction. 
Judging by the brilliance of the image of the candle when viewed 
through the Nicol at nearly normal incidence, and distances at 



v 


































Q 


































Q| 


8 
7 
6 
5 
4 
3 
2 
1 

( 


































// 


































M 


































n\\ 
































/ 


l\\ 
































/ 


Ml 






























i 




\ \ 






























/ 




1 II 1 




























/ 




/ 




























s. 


/ 




/ 


























r$ 


y 




*>/ 




























> 




& 


/ 
























s 


/ 




^ 


r 






i 


















^ 






/ 


/ 













=3 


=S 


. ' 

=: 


. * 
_^ 


^ 

. H 

*-*, 


' 
-\, 


, ' 


- 


^* 


^ 










j 


3* I 


















^ 


^ 












/ 
























^ 


^, 






^ 






) 10 20 30 40 60 60 70 80 9O* 



Fig. 0. Diamond. 



2-434) 



which this image can be seen, I think we may safely guess that the 
light reflected from the water at normal incidence was at least 500 
(instead of Green's 3J) times as strong as the imperceptible light 
of the nearest approach to extinction which the Nicol gave at the 
polarizing angle of incidence. And from Rayleigh's accurate experi- 
ments ( 105 below) we know that it is 25,000,000 times, when the 
water surface is uncontaminated by oil or scum or impurity of any 



348 



LECTURE XVTTT. 



Molar, kind. It would he only 34- times if Green's dynamics were applicable 
(without change of hypotheses) to the physical problem. Hence, 
looking at fig. 7, we see that, for water, Green's theory (Curve 3) 
differs enormously from the truth, while Fresnel's formula 
(Curve 2) shows perfect agreement with the truth, at the angle of 
incidence 53 C< 1. Looking at fig. 4 we see still greater differences 



1-0 



-9 



\ 



\\ 



\ / 



10 20 30 40 50* 60 70 8O 90 

Fig. 7. Water. 

between Curves 2 and 3 even at as high angles of incidence as 
75, though there is essentially a perfect concurrence at 90. 

105. Looking at Curve 3 in figs. 5 and 6 we see that for flint 
glass Green's theory gives scarcely any diminution of reflected 
light, while for diamond it actually gives increase, when the angle 
of incidence is increased from zero to tan~>, Brewster's polarizing 



FRESNEL, GREEN, AND TRUTH. 349 

angle. Yet even a hasty observation with no other apparatus Molar, 
than a single Nicol's prism shows, both for flint glass and diamond, 
diminution from the brightness of directly reflected light to what 
seems almost absolute blackness at incidence tan -1 /i, when the 
reflection is viewed through the Nicol with its plane of polariza- 
tion perpendicular to the plane of reflection (that is to say, its 
transmitted light having vibration in the plane of reflection). 

One readily and easily observed phenomenon relating to 
polarization by reflection is: with a Nicol tunied so as to transmit 
the vibrations in the plane of reflection, diminution of reflected 
light to nearly zero with angle of incidence increasing to tan" 1 /*; 
and after that increase of the reflected light to totality when the 
angle of incidence is farther increased to 90. 

Another is: for light reflected at a constant angle of incidence, 
diminution from maximum to minimum when the Nicol is turned 
90 round its axis so as to bring the direction of vibration of the 
light transmitted by it from being perpendicular to the plane of 
reflection to being in this plane. This diminution from maximum 
to minimum is the difference between the ordinates of two curves 
representing, respectively for vibrations perpendicular to the plane 
of incidence and vibrations in the plane of incidence, the intensity 
of the reflected light at all angles of incidence. These two curves, 
if drawn with absolute accuracy, would in all probability agree 
almost perfectly with Fresnel's sine-law and Fresnel's tangent-law 
(Curves 1 and 2 of the diagrams) for all transparent substances. 
We have, however, little or no accurately measured observational 
comparisons except for light incident at very nearly the polariza- 
tional angle. By a very different mode of experimenting from 
that indicated in 95 above, Jamin* found for eight substances 
having refractive indices from 2'454 to 1'334, results shown in 
the following table ; to which is added a measurement made by 
Rayleigh for water with its surface carefully purified of oil or 
scum or any other substance than water and air. 

* Ann. de Chimie et de Physique, 1850, VoL xxix., p. 303, and 1851, Vol. xxxi., 
p. 179 ; and p. 174 corrected by pp. 180, 181. (Confusion between Cauchy's e and 
Jamin's k is most bewildering in Jamin's papers. After much time sadly spent in 
trying to find what was intended in mutually contradictory Tables of results, 
I have given what seems to me probably a correct statement in the Table belonging 
to 105.) 



350 



LECTURE XVIII. 



Substance 


M 


Angle of 
Incidence 

tan" 1 /u. 


Ratios of vibrational ampli- 
tudes, and of strengths, of 
reflected lights due to equal 
incident lights vibrating in 
and perpendicular to the 
plane of incidence 


"Sulfure d'arsenic trans- 
parent " (Realgar) . . . 
"Blende transparente " 
(zinc sulphide) 


2-454 

2-371 
2-434 
1-714 
1-487 
1-441 
1-366 
1-334 

1-334 


67'8 

67-l 

67-7 
59-7 
56-l 
55-2 
53-8 
53-l 

53-l 


K 

+ -0850 

+ -0420 
+ 0190 
+ -0180 
+ -0060 
- -0084 
+ -00208 
- -00577 

+ -0002 


V 
1/138 

1/567 
1/2770 
1/3086 
1/27778 
1/14172 
1/231160 
1/30030 

1/25,000,000 


Diamond 


Flint Glass 


" Verre " 


Fluorine 


Absolute Alcohol 
Water (Jamin) 


Water, with specially puri- 
fied surface (Rayleigh) 



Molar. The greatest numeric in the last column, 1/138, would be barely 
perceptible on the diagram, fig. 7 ; and none of those for the other 
seven substances would be perceptible at all without a large 
magnification of the scale of ordinates. 

106. Although these results are related only to the ratio of 
the ordinates of Curve 1 to those of Curve 2 for one angle of 
incidence in each case, and do not touch the absolute values of the 
reflection due to unit quantities of incident light, we may infer as 
almost absolutely certain, or at all events ( 100, 125) highly 
probable, that Curve 1 (Fresnel's sine-law) and Curve 2 (Fresnel's 
tangent-law) are each of them about as nearly correct at all other 
incidences as at the critical incidences for which the observations 
were made. We shall see in fact ( 125, 133 below) in the dy- 
namical theory to which we proceed, that the sine-law is absolutely 
accurate for vibrations perpendicular to the plane of incidence, on 
the supposition that the rigidities of the two mediums are equal 
and their densities unequal ; and that the only correction of Green's 
dynamical postulates which can procure approximate annulment, 
for the reflected light, of vibrations in the plane of incidence at 
the angle tari" 1 /!, gives correspondingly close agreement with 
Fresnel's tangent-law throughout the whole range of incidences 
from to 90 I 



DYNAMICAL THEOKY OF REFLECTION AND REFRACTION. 351 

(The following, 107...111, is quoted from a paper written Molar, 
by myself in September October, 1888, and published in the 
Philosophical Magazine for 1888, second half year.) 

107. " Since the first publication of Cauchy's work on the 
" subject in 1830, and of Green's in 1837, many attempts have 
" been made by many workers to find a dynamical foundation 
" for Fresnel's laws of reflection and refraction of light, but all 
" hitherto ineffectually. On resuming my own efforts since the 
"meeting of the British Association at Bath, I first ascertained 
"that an inviscid fluid permeating among pores of an incora- 
" pressible, but otherwise sponge-like, solid does not diminish, but 
"on the contrary augments, the deviation from Fresnel's law of 
"reflection for vibrations tn the plane of incidence. Having 
" thus, after a great variety of previous efforts which had been 
"commenced in connexion with preparations for my Baltimore 
'" Lectures of this time four years ago, seemingly exhausted 
"possibilities in respect to incompressible elastic solid, without 
"losing faith either in light or in dynamics, and knowing that 
" the condensational -rarefactional wave disqualifies* any solid of 
" positive compressibility, I saw that nothing was left but a solid 
"of such negative compressibility as should make the velocity of 
" the condensatioual-rarefactional wave, zero or small. So I tried 
" it and immediately found that, with other suppositions unaltered 
fr<in Green's, it exactly fulfils Fresnel's 'tangent-law' for vibra- 
" tions in the plane of incidence, and his ' sine-law ' for vibrations 
"perpendicular to the plane of incidence. I then noticed that 
" homogeneous air-less foam, held from collapse by adhesion to 
"a containing vessel, which may be infinitely distant all round, 
" exactly fulfils the condition of zero velocity for the condensational- 
" rarefactional wave ; while it has a definite rigidity and elasticity 
"of form, and a definite velocity of distortional wave, which 
" can be easily calculated with a fair approximation to absolute 
" accuracy. 

108. " Green, in his original paper ' On the Reflexion and 
"Refraction of Light' had pointed out that the condensational- 
" rarefactional wave might be got quit of in two ways, (1) by its 
" velocity being infinitely small, (2) by its velocity being infinitely 

Green's Collected Papert, p. 246. 



352 LECTURE XVIII. 

Molar. " great. But he curtly dismissed the former and adopted the 
"latter, in the following statement: 'And it is not difficult to 
" ' prove that the equilibrium of our medium would be unstable 
'"unless A/B>4:/3. We are therefore compelled to adopt the 
'"latter value of A/B*,' (GO) 'and thus to admit that in the 
" ' luminiferous ether, the velocity of transmission of waves pro- 
"'pagated by normal vibrations is very great compared with that 
"'of ordinary light.' Thus originated the 'jelly-' theory of ether 
" which has held the field for fifty years against all dynamical 
" assailants, and yet has hitherto failed to make good its own 
" foundation. 

109. " But let us scrutinize Green's remark about instability. 
" Every possible infinitesimal motion of the medium is, in the 
" elementary dynamics of the subject, proved to be resolvable into 
"coexistent equi-voluminal wave-motions, and condensational- 
" rarefactional wave -motions. Surely, then, if there is a real 
" finite propagational velocity for each of the two kinds of wave- 
" motion, the equilibrium must be stable ! And so I find Green's 
" own formula "f* proves it to be provided we either suppose the 
"medium to extend all through boundless space, or give it a fixed 
" containing vessel as its boundary. A finite portion of Green's 
" homogeneous medium left to itself in space will have the same 
" kind of stability or instability according as A JB > 4/3, or 
" A/B < 4/3. In fact A $B, in Green's notation, is what I have 
"called the 'bulk-modulus'* of elasticity, and denoted by k 
" (being infinitesimal change of pressure divided by infinitesimal 
" change from unit volume produced by it : or the reciprocal of 
"what is commonly called 'the compressibility'). B is what 
" I have called the ' rigidity,' as an abbreviation for ' rigidity- 
" modulus,' and which we must regard as essentially positive. 
" Thus Green's limit A/B > 4/3 simply means positive compres- 
sibility, or positive bulk-modulus: and the kind of instability 
"that deterred him from admitting any supposition of A/B< 4/3, 
" is the spontaneous shrinkage of a finite portion if left to itself 

* A and B are the squares of velocities of the condensational and distortioual 
waves respectively ; supposing for a moment the density of the medium unity. 

t Collected Papers, p. 253 ; formula (C). 

Encyclopaedia Britannica, Article " Elasticity" : reproduced in Vol. in. of my 
Collected Papers. 



VOLUMINAL INSTABILITY AND STABILITY. 353 

"in a volume infinitesimally less, or spontaneous expansion if Molar. 
"left to itself in a volume infinitesimally greater, than the 
" volume for equilibrium. This instability is, in virtue of the 
"rigidity of the medium, converted into stability by attaching 
" the bounding surface of the medium to a rigid containing vessel. 
" How much smaller than 4/3 may A/B be, we now proceed to 
" investigate, and we shall find, as we have anticipated, that for 
" stability it is only necessary that A be positive. 

110. "Taking Green's formula (C); but to make clearer the 
" energy-principle which it expresses (he had not even the words 
"'energy,' or 'work'!), let W denote the quantity of work re- 
" quired per unit volume of the substance, to bring it from its 
"unstressed equilibrium to a condition of equilibrium in which 
" the matter which was at (ac, y, z) is at (x + , y + rj, z + ) ; 
" ? f being functions of x, y, z such that each of the nine 
"differential coefficients d^/dx, d/dy, ... dtjldx... etc. is an in- 
" finitely small numeric; we have 

dri d ?' 



dr)\\ 
Ty)\ 



d? d 
y dz + d 

"This, except difference of notation, is the same as the formula 
"for energy given in Thomson and Tait's Natural Philosophy, 
"693(7). 



111. "To find the total work required to alter the given 
"portion of solid from unstrained equilibrium to the strained 

"condition (f, 17, 5") we must take \\\dxdydzW throughout the 

"rigid containing vessel Taking first the last line of (1); 
"integrating the three terms each twice successively by parts 
"in the well-known manner, subject to the condition =0, 17 = 0, 
" f = at the boundary ; we transform the factor within the last 
" vinculum to 



T. L. 



354 LECTURE XVIII. 

Molar. " Adding this with its factor 45 to the other terras of (1) 
under llldxdydz, we find finally 



(2). 



" This shows that positive work is needed to bring the solid to 
" the condition (, 77, ) from its unstrained equilibrium, and 
" therefore its unstrained equilibrium is stable, if A and B are 
" both positive, however small be either of them." 

112. The equations of motion of the general elastic solid 
taken direct from the equations of equilibrium, with p to denote 
density, are, as we found in Lecture II. pp. 25, 26 



^f = dP dU dT 
p dt* ~ dx + dy + dz 

U dQ dS 



dT dS dR 
dx dy dz ) 



-(3); 



where , tj, denote (as above from Green) displacements ; P, Q, R, 
normal components of pull (per unit area) on interfaces respec- 
tively perpendicular to x, y, z; and S, T, U respectively the 
tangential components of pull as follows : 

o f = pull parallel to y on face perpendicular to z 

z y 



U 



,.(4). 



113. For an isotropic solid we had in Lecture XIV. (p. 191, 
above), 



GENERAL SOLUTION FOR MOTION OF ELASTIC SOLID. 355 






where 5 = ^| + ^? + ^ (7). 

Using these values of S, T, U, P, Q, R in (3) we find 

(8). 

114. Taking djdx of the first of equations (8), d/dy of the 
second, and d/dz of the third, and adding we find 

where A = fc + fn (10), 

this being Green's "A" as used in 108, 111 above. 
Put now 

which implies f^ + ^r + ^r-O ( 12 > 

cue du dz 

and we find, by (8), 

*&_-.,!. -<&ll 



Equations (9), (12), and (13) prove that any infinitesimal dis- 
turbance whatever is composed of specimens of the condensa- 
tional-rarefactional wave (9), and specimens of the distortional 
wave (13), coexisting; and they prove that the displacement in 
the condensational-rarefactional wave is irrotational, because we 
see by (11) that an absolutely general expression for its com- 
ponents, , T) rj lt f &t ^ denoted by 2 , 7/ 2 , 2 , is 
dV cW dV 



232 



356 LECTURE XVIII. 

Molar, where, when 8 is known, "^ is determined by 



S* ........................... (15). 

Hence, as 8 satisfies (9), we have 

fJVfy 

p-AV*9 ...................... (16); 

and we see, finally, that the most general solution of the equations 
of infinitesimal motions is given by 



, 1., 

provided , rj l , satisfy (12) and (13); and 2 > fy, 2 satisfy (14) 
and (16). 

115. The general solutions of (11) and (12) for plane equi- 
voluminal waves, and of (14) and (16) for plane condensational - 
rarefactional waves are as follows (easily proved by differentia- 
tions): 



with 



(19); 

with 

P 

where H is a constant, equal to the displacement in the con- 
densational-rare factional wave when f= 1 ; A, B, C, are constants 
equal to the x-, y-, ^-components of the displacement, due to 
the equivoluminal wave when f= ; (a 1} fti, 71), (a 2 , /9 2 , 72) are 
the direction-cosines of the normals to the wave-planes f of the 

* Poisson's well-known fundamental theorem, in the elementary mathematics 
of force varying inversely as the square of the distance, tells us that when 5 is 
known, or given arbitrarily through all space, V~ 2 5 is determinate; being the 

potential of an ideal distribution of matter, of which the density is equal to . 

t By "wave-plane" of a plane wave I mean any plane passing through particles 
all in one phase of motion. For example, in sinusoidal plane waves the "wave- 
plane " may be taken as one of the planes containing particles having no displace- 
ment but maximum velocity, or it may be taken as one of the planes having 
maximum displacement and no velocity. For an arbitrary impulsive wave (as 
expressed in the text with / an arbitrary function through a finite range, and zero 
for all values of the argument on either side of that range) the " wave-plane " may 



- 1 - - ft - 

= = = 




REFLECTION AND REFRACTION AT INTERFACE BETWEEN SOLIDS. 357 



two waves ; and u, v are the propagational velocities of the equi- Molar. 
voluminal and condensational-rarefactional waves respectively. In 
the condensational-rarefactional wave in an isotropic medium, the 
displacement or line of vibration is in the direction (a,, $ 7,), 
normal to the wave-plane. 

116. For the problem of reflection and refraction at an 
interface between two mediums, which (following Green) we shall 
call the upper medium and the lower medium respectively, let the 
interface be XOZ, and let this plane be horizontal. Let the 
wave-planes be perpendicular to XOY. This makes 71 = 0, 7j = 
in (18) and (19). For brevity we shall frequently denote by P, 
the plane of incidence and reflection. 

117. Beginning now with vibrations perpendicular to P, 
we have .4 = 0, B = 0; and (18) becomes, for an incident wave as 
represented in fig. 8, 

S=Cf(t-ax + by) ..................... (20), 

where 

a = sin i/u, and b = cos ifu, with u = /- ...... (21). 

V p 



O 



r 



p g-j 



Fig. 8. 



OZ is perpendicular to the diagram towards the eye. 

The wave-planes of incident (/), reflected (/'), and refracted 
(,/), waves are shown in fig. 8 for the particular case of inci- 
dence at 30, and refractive-index for flint-glass = 1 '7 24; which 
makes ,t = 16'9. 

be taken as any plane through particles all in the same phase of motion. In this 
case we have a wave-front and a wave-rear; in the sinusoidal wave we have no 
front and no rear. I have therefore introduced the word "wave-plane" in prefer- 
ence to the generally used word " wave-front." 



358 LECTURE XVIII. 

Molar. For the reflected and refracted waves we may take respectively 
180 i, aud ,i instead of i, and C", f C instead of G. We have 
sin t i sin i 



,, with ' M = v/'^ < 22 >< 

and C' and ,0 by equations (28) below. Thus for the displace- 
ments in the two mediums, due to the three waves, we have 
+'= Cf(t ax + by) + C'J(t axby)in upper medium... (23), 
and 

=,Cf(t ax + ,by) in lower medium... (24), 

where ,b = cos ,i/,w, with ,u = */ (25) ; 

,6 and t u being both positive. Remember that y is negative in 
the lower medium. Remark that, by (21), (22), (25), we have 

b* = u- 2 -a 2 , ,b* = ,u-*-a* (25)'. 

The sole geometrical condition to be fulfilled at the interface is 

+'=, when y=0, which gives C + C' = ,G ...(26). 
The sole dynamical condition is found by looking to 113 (5). It is 

S=,S when y = ; giving nb (G - C') = ,n,b,G ...(27). 

From these we find 

c , = bn-,bn 2bn 

,b,n + bn ' ,b,n + bn 

118. The interpretation of these formulas is obvious when 
the quantities denoted by the several symbols are all real. But in 
an important and highly interesting case of a real incident wave, 
expressed by the first term of (23) with all the symbols real, 
imaginaries enter into (24) and (25) by ,b being imaginary ; 
which it is when 

a- 1 <,u (29); 

or, in words, when the velocity of the trace of the wave-planes on 
the interface is less than the velocity of the wave in the lower 
medium. In this case a 2 t u~ 2 is positive: denoting its value by q* 
we may put ,b = iq*, where q is real; positive to suit notations 
in 119. Thus (28) becomes 

n , bn + iq,n n b 2 n 2 q 2 ,n 2 + ^tqbn^i ~ 

C/ = j = 7 O 

on iq,n b z n 2 + q 2 t n 2 

c _ 2bn (bn + iq,n) 



bn iq,n 

* See foot-note on 158" below. 



TOTAL REFLECTION OF EQUI-VOLUMINAL WAVES. 359 

We have also in (24) an imaginary, ,b = 2q, in the argument of /Molar. 
for the lower medium. 

119. To get real results we must choose / conveniently to 
make f(t ax iqy) = F + tO, where F and G are real. We may 
do this readily in two ways, (31), or (32), by taking T, an arbitrary 
length of time, and putting 

1 t 

- ( _- -j-j-j- _ ( 

This makes 



t-aa:-2qy + 2r (t - ax? + (T - qy? 
or 

f(t ax+by) = e'<'-*+&v> _ C os<o( ax+by) + 1 sin <u (t ax + by)] 
f(t - ax+,by) =---w) = ew [cos o>(t - ax) + 1 sin <w (t - ax)] j 

......... (32). 

The latter of these is the proper method to show the results 
following the incidence of a train of sinusoidal waves ; the former, 
which we shall take first, is convenient for the results following 
the incidence of a single pulse. 

120. With (23), (24), and (30), it gives for the incident wave 
in the upper medium 

t-ax + by-tr r 
= C .................. (33) ' 



and for the reflected wave in the upper medium 

- ax-by)+2qbn,riT]+i [2qbn,n (t-ax- 6y)-(feV-g*,n a 



(6*n + 3 > 8 ) [(t-ax- by? + r 2 ] 

......... (34), 

and for the disturbance in the lower medium 

2 j[6V (t - ax) + qbn,n (r - qy)] + 1 [qbn,n (t-ax)- 6V (r - qy)]} 
(V> n > + q* >n >)[(t-ax? + (T-qyy>] 

......... (35). 

The real parts of these three formulas represent a certain form 
of arbitrarily given incident wave : and the consequent reflected 
wave in the upper medium, and disturbance (a surface-wave, 



H60 LECTURE XVIII. 

Molar, analogous to a forced sea-wave) in the lower medium. The 
imaginary parts with i removed, represent another form of incident 
wave and its consequences in the upper and lower mediums. In 
neither case does any wave travel into the lower medium away 
from the interface, and therefore the whole activity of the incident 
wave is in each case carried on by the reflected wave in the upper 
medium ; that is to say, we have total reflection. It is interesting 
to see that in this total reflection, the reflected wave in each case 
differs in character from the incident wave, except for direct 
incidence ; and it differs by being compounded of two constituents, 
one of the same character as the incident wave for that case, 
and the other of the same character as the incident wave for 
UK; other case. This corresponds to the change of phase ( 152 
and 158 |V , below) by total internal reflection of waves of vibration 
perpendicular to P. 

1 21. The accompanying diagram (fig. 9) shows the characters 
of the two forms of incident wave, and of two constituents of the 
forced surface-wave in the lower medium, referred to in 120. 
The two curves represent, from t = <x> to t = + <x> , the part of 
the displacement of any particle in the upper medium, due to 
one or other alone of the two forms of incident wave. The 
abscissa in each curve is t. The ordinates of the two curves 
are as follows : 

Curve 1, , , and Curve 2, - 

P + T J P + T 2 

The unit of ordinates in each case is T" 1 . 

The disturbance in the lower medium is a forced wave, of 
character represented by a combination of these two curves, 
travelling under the interface at speed a" 1 . 

122. All the words of 120 apply also to the total reflection 
of sinusoidal waves, with this qualification, that the two char- 
acters of incident wave are expressed respectively by a cosine 
and a sine, and the " difference " becomes simply a difference of 
phase. Thus having taken the real parts of the formulas we 
get nothing new by taking the imaginary parts. The real parts 
of (23) and (24), with (.'12) for / and with (30) for C' and ,C, 
give us 

Incident wave, C cos to (t ax + by) (36); 



TOTAL REFLECTION OF EQUI-VOLUMINAL PULSES. 361 









362 LECTURE XVIII. 

Molar. /'Reflected wave, 

< (b 2 n z -q z ,n*)cost0(t-ax-by 
( & 2 ?i 2 +2 

(Forced wave in lower medium, } 

/oo\ 
2bn e-^w [bn cos w (t ax} q,n sin w (t ax)] ? ' 
bW + q*,n 2 J 

Thus we see that the amplitude of the resultant reflected wave 
is C; that its phase is put forward tan" 1 [2qbn,n/(ljPn* <? 2 ,n 2 )] ; 
and that the phase of the forced wave-train in the lower medium 
is before that of the incident wave by tan" 1 (q,n/bn). 

123. Leaving for 158', the case of total reflection, and 
returning to reflection and refraction, that is to say, b and ,b both 
real, we see that equations (28) give for the case of equal rigidities 

C' _ ,b b _ cot ,i cot i _ sin (i ,i} 
C ~ ,b + b ~~ cot ,i + cot i ~ sin (i + ,i) ' ' 

which is Fresnel's "sine-law"; and, belonging to it for the 
refracted ray, 

,G _ 2 cot i _ 2 cos i sin ,i 

C cot ,i -f cot i sin (,i + i) 

In the case of equal densities and unequal rigidities we have 
n/,n = sin 2 ?'/sin 2 ,i and equations (28) give 

cos i sin ,i 



i sin i _ sin 2i sin 2,t _ tan (i ,i) 

~ ~ " ' 



_ cos , 

C ~ cos i sin ,i ~ sin 2i + sin 2,i ~ tan (i + ,i 
cos ,i sin i 

which is Fresnel's "tangent-law"; and, belonging to it for the 
refracted ray, 

,C _ 2 sin i cos i sin 2t 

C ~ sin ,i cos ,i + sin i cos * ~ sin 2,i + sin 2i " ' 

The third member of (41) is in some respects more convenient 
than Fresnel's beautiful " tangent-formula." 

124. These formulas, valid for pulses as well as for trains 
of sinusoidal waves, show, without any hypothesis in respect to 
compressibility or non-compressibility of ether, that, if the densities 



DISPLACEMENTS PERPENDICULAR TO PLANE OF INCIDENCE. 363 

of the mediums are equal on the two sides of the interface, Molar. 
Fresnel's "tangent-law" is fulfilled by the reflected waves, if the 
vibrations are parallel to the interface ; and therefore the reflected 
light vanishes when the angle of refraction is the complement of 
the angle of incidence. Hence non-polarized light incident at 
the angle which fulfils this condition would give reflected light 
consisting of vibrations in the plane of the incident and reflected 
rays. Now ( 81) we have seen from Stokes' dynamical theory of 
the scattering of light from particles small in comparison with 
the wave-length, as in the blue sky, and also from his grating 
experiments and their theory, and ( 81') confirmation by Rayleigh 
and by Lorenz of Denmark, that, in light polarized by reflection, 
the vibrations are perpendicular to the plane of polarization 
defined as the plane of the incident and reflected rays; that is to 
say the vibrations in the reflected ray are parallel to the interface. 
Hence it is certain that the densities of the mediums are not 
equal on the two sides of the interface ; and that the densities 
and rigidities must be such as not to give evanescent reflected ray 
for any angle of incidence, when, as in 123, the vibrations are 
parallel to the interface. 

125. On the other hand, formulas (39), (40) show that the 
supposition of equal rigidities not only does not give evanescent 
ray for any angle of incidence, but actually fulfils Fresnel's 
"sine-law" of reflection for all angles of incidence when the vibra- 
tions are parallel to the interface. Looking back to (28) and (39), 
we see that Fresnel's "sine-law" is expressed algebraically by 

,6,n - bn _ ,b-b ,. 

,b,n+bn~,b + b~ 

If this equation is true for any one value of &/,& other than 
or , we must have n/,n = 1. Hence if, with vibrations parallel 
to the interface, Fresnel's "sine-law" is exactly true for any one 
angle of incidence other than 90, the rigidities of the mediums 
on the two sides are exactly equal, and the "sine-law" is exactly 
true for all angles of incidence ; a very important theorem. 

126. Going back now to the more difficult case of vibrations 
in the plane of incidence, let us still take the interface as XOZ 
and horizontal; and the wave-planes perpendicular to XOY. 
Instead of the single displacement-component, and the single 



364 LECTURE XVIII. 

Molar, surface-pull-component, S, of 117 125, we have now two dis- 
placement-components, , 77 ; and two surface-pull-components, Q, 
perpendicular to the interface, and U parallel to the direction X 
in the interface. The interfacial conditions are that each of these 
four quantities has equal values on the two sides of the inter- 
face. We have now essentially the further complication of two 
sets of waves, equivoluminal and condensational-rarefactional, 
which are essentially to be dealt with. We might suppose the 
incident waves in the upper medium to be simply equivoluminal 
or simply condensational-rarefactional ; but incidence on the inter- 
face between two mediums of different densities or different 
rigidities would give rise to reflected waves of both classes in the 
upper medium, and to refracted waves of both classes in the 
lower medium. It will therefore be convenient to begin with 
incident waves of both classes in the upper medium. 

127. Going back now to 115, according to the details chosen 
in 126, and taking j for the angle of incidence of condensational- 
rarefactional waves, we have % = 0, (7=0, j 2 = ; and we may put 

i = sin i, A = G cos i. ou = sin j, ) 

i ^44^ 

fa = - cos j.j ' 



Thus instead of (18) and (19) we now have 

x sin i y cos i 



cos i sin i \ u 

with 



n 

U = A/- 

V p 



-*- = Hf(t- <^J^y^J] 

C SJ J \ (46), 

with "Vn? 12 

as the two constituents for the incident waves. Let now ,u, ,G, ,i, 
and ,v, ,H, ,j be the values of the constants for the waves in the 
lower medium. 

For the reflected waves in the upper medium we have cos i, 
cosj, instead of cos i, cosj; while sin i, sinj are the same as for 
the incident waves. Let - G', + H', be the constants for the 
magnitudes of the reflected rays corresponding to G, H, for the 



DISPLACEMENTS IS PLANE OF INCIDENCE. 365 

incident rays. Thus for the total displacement-components due 
to the four waves Jn the upper medium we have 






and for total displacement-components in the lower medium the 
same formulas with ,, J, ,, p, ,G, JT, 0, 0, in place of i, j, u, v, 
G, H, G', H" respectively. 

128. Taking from these formulas the resultants of the Cf 17) 
components of the displacements in the several waves, we have as 
follows: 



Eqaivoluminal wave 



Reflected, 



129. Patting now y = to find displacement-components at 
the interface, and equating values on the two sides, we see in the 
first place that the arguments of/ with y zero must be all equal ; 
and that the coefficient of x is the reciprocal of the velocity of the 
trace of each of the four waves on the interface ; and if we denote 
this velocity by or 1 we have 

nj = smj 



366 LECTURE XVIII. 

Molar. The last three of these equations express the laws of refraction 
of waves of either class in the lower medium consequent on waves 
of either class in the upper medium. They show that the sines of 
the angles of incidence and refraction are inversely as the propa- 
gational velocities in the two mediums, whether the two waves 
considered are of the same species or of the two different species. 
For brevity in what follows we shall put 

cos* 7 cos, i 7 cos j cos, j /AC .. 

~=b, =,b, ~=c, ^ = ,0 (49), 

u ,u v ,v 



from which we find 

*-. " 8 +^-=^k>) 

...(50). 




a 2 + ,6 2 = ,w~ 2 = ^ ; a 2 + ,c 2 = ,*r 2 
130. Putting now f(t ax) = -/r, we find by (47) with y = 



rj = [au (G-G') + cv(-H + H')] 

as the displacement-components of the upper medium at the 
interface. 

Using now (47) to find Q and U by (6) and (5) of 113, and 
putting y = we find, as the components of surface-pull of the 
upper medium at the interface 

Q = n [2dbu (G + G') - hv (H + H')] f 1 _ 



where ff = b 2 a 2 , 

Taking the terms of (51) and (52) which contain G, and H, 
and in them substituting ,G, ,H, ,b, ,c, ,g, ,h, ,v, ,n, ,k for G, H, b, c, 
g, h, v, n, k, we find for the two refracted waves in the lower 
medium the displacement-components, and the surface-pull-com- 
ponents, at the interface, as follows : 



.(54). 



SOLUTION FOR DISPLACEMENTS IN PLANE OF INCIDENCE. 367 

Equating each component for the two sides, as said in 126, Molar. 
we find the following four equations for determining the four 
required quantities, ,0, ,H, G', H', in terms of the two given 
quantities 0, H, 



bu(G+G') + av(H + H') = ,b,u,G+ a,v,H 

au(0-G") -cv(H-H')= a,u,G- ,c,v,H 

n[2abu(G + G') -hv(H+ H')} = ,n[Za,b,u,G- ,A,vW' 
n\gu(G- G') + 2acv (H-H')] = ,n (,g,u, 



131. Considering for the present G G' and H + H' as the 
known quantities, and G + G', H-H', ,G, ,H, as unknown; first 
find two values of G + G' from the first and third of equations (55), 
and two values of H H' from the second and fourth of (55). 
Thus we have 



+ 0' 






The equalities of the second and third members of these two 
double equations may be taken as two equations for the deter- 
mination of ,G, ,H in terms of G + G', and H + H'. With some 
simplifying reductions they become 

2a,6(n - ,),, + [# + 2(n - ,n)a],t;,J5r = pv (H+H')\ 
lp + 2(n - ,n) a] ,u,G - 2a,c (n - ,n) ,v,H = pu(G- G')]" 

Finding ,G and ,H from these, and using the results in either 
the firsts or the seconds of the pairs of equations (56), we have 
G + G' and H - H' in terms of G - G' and H + H'. These last 
two equations may be used to find G' and H' in terms of G and H. 
Thus we have finally G', H', ,G, ,H in terms of the given 
quantities G, H. This, of course, might have been found directly 
from the four equations (55) by forming the proper determinants. 
The algebraic work is somewhat long either way. 



368 LECTURE XVIII. 

Molar. | 132. But the process we have followed in 131 has the 
advantage of giving us the two intermediate equations (57); 
with the great simplification which they present in the case 
n = ,n, for which they become 

,p,v,H=pv(H + H'), ,p,u,G = pu(G-G') (58). 

Let us now work this case out to the end, with the further 
simplification H=0; because the particular problem which we 
wish to solve is to find the two reflected waves (G', H') and the 
two refracted waves (,G, ,H) due to a single incident equi- 
voluminal wave (G). Eliminating t G, ,H from the first two of (55), 
by (58) with H=0; and then finding G' and H' from the two 
equations so got, we have 

.pb-pfi-L 



,pb- 

L 



where L= c c < 61 )- 

Lastly (58) gives ,G= 2 - pfe ^ G (62), 

P*' z ' 



v 



(63). 



This completes the theory of the reflection and refraction of 
waves at a plane interface of slipless contact between two ordinary 
elastic solids, with any given bulk-moduluses and rigidities. 






133. Remark now that by equations (49), (48), we have 
,6 t< cos ,i _ sin i cos ,i 
,u cos i ~" sin ,i cos t 
and by (50) with n = ,n, p/,p = ,u*/u* (65); 

p,b sin .i cos .i 

whence, by (64), '- (66). 

,pb sin i cos t 

Hence, we see that if L = we have 

G' _ sin 2t sin 2,i _ tan (i ,i) ^ 

~G ~ sin 2i + sin 2,i = tan (i + ,i) ' '* 

which is Fresnel's formula for reflection when the vibrations are in 
the plane of incidence. 



TRIAL OF UNEQUAL RIGIDITIES. 369 

Looking to (61) we see that this could only be the case for Molar. 
other than direct incidence by either c, or ,c, or both being 
infinitely great; because, 129 (48), a can only be zero for direct 
incidence (t = 0). By (50) we see that to make c or ,c very great, 
we must have v or ,v very small. Returning to this suggestion in 
(Lect. XIX. 167), we shall find good physical reason to leave, for 
undisturbed ether, v = oo as Green made it : and to let ,v be small 
enough, to make L not absolutely zero but as small as required 
to give the closeness of approximation to truth which observation 
proves for Fresnel's formulas for the great majority of transparent 
liquids and solids, including optically isotropic crystals; and yet 
large enough to give modified formulas representing the devia- 
tions from Fresnel found in diamond, sulphide of zinc, sulphide of 
arsenic, etc. (see 105 above and 182 below). 

134. Meantime we shall consider some interesting and 
important characteristics of the general problem of 130 without 
the limitation n = ,n : and of its solution for the case n = ,n 
expressed by 123 (39), (40) .and 132 (59).. .(63), with the 
modification due to v = oo in space containing no ponderable 
matter ; and with the special further modification regarding waves 
or vibrations of ether in the space occupied by metals (solid or 
liquid) to account for the observational ly discovered facts of 
metallic reflection. 

135. Consider first direct incidence, whether of an equi- 
voluminal or of a condensational-rarefactional wave. For this we 
have in 129, 130, 



.(68). 



These details reduce (55) to 

G + G' = ,0 

H-H'= ,H 

pv(H + H')=,p,v f H 

pu(G-G') = ,p,u,G 

The first and fourth of these give G' and ,G in terms of G ; the 
T.I, 24 



370 LECTURE XVIII. 

Molar, second and third give H' and ,H'vn. terms of H. Thus our formulas 
verify what is obvious without them ; that a directly incident wave, 
if equivoluminal, gives equivoluminal reflected and transmitted 
waves ; and if condensational-rarefactional, gives condensational- 
rare factional reflected and transmitted waves. For direct incidence, 
remark that in the equivoluminal waves the displacement is parallel 
to the reflecting surface, and in the x direction ; in the conden- 
sational-rarefactional waves it is perpendicular to the reflecting 
surface, which is the y direction. For ratios of these displace- 
ments we find, as follows, from (68) ; 

, . . . .. G-' .p.u pu ,G 2pu 

(Equivoluminal) Vr -=- -; - ......... (69); 

i G 



i -x H' >P, V pv iH 2pv 

(Condensational-rarefactional) f7 - = - , -77 

H v + v H 



...... (70). 

In respect to the general theory of the reflection of waves at 
a plane interface between two elastic solids of different quality, 
it is interesting to see that, provided only they are sliplessly 
connected at the interface, we have, for the case of direct incidence, 
the same relations between the displacements of the reflected and 
transmitted waves and of the incident wave in terms of densities 
and propagational velocities, for equivoluminal waves of transverse 
vibration, as for condensational-rarefactional waves (vibrations in 
the line of transmission). If, on the other hand, the connection 
between the two solids were merely by normal pressure, and if 
the surfaces of the two solids at the interface were perfectly 
Motionless, and allowed perfect freedom for tangential slipping ; 
the reflection in the case of directly incident waves of transverse 
vibration would be total, and there would be no transmission of 
waves into the other solid. 

136. In respect to physical optics the solution (69) is ex- 
ceedingly interesting. If we eliminate p, ,p from it by 



we find 

G'_ ,nu n,u ,G _ 2n,w .^ . 

G .,nu + n,u ' G ,nu 4- n,u 

Hence, denoting u/,u by //, (the refractive index), we see that, 



DIRECT REFLECTIVITY PROVES EQUAL RIGIDITIES OR DENSITIES. 371 

for the reflected wave, in the two cases of equal rigidities and Molar, 
equal densities, we have respectively, 



Thus G'/G is equal but with opposite signs in the two cases; 
and therefore, as the ratio of the intensity of the reflected light 
to the intensity of the incident light is equal to (G'/GJ*, we see 
that it is equal to 

...(73), 



and is the same in the two cases of equal densities and equal 
rigidities; an old known and very important result in physical 
optics. It was, I believe, first given by Thomas Young; it is 
also found by making t = in Fresnel's formulas for reflection of 
polarized light at any incidence. But full dynamical theory 
proves that, if the refractivity p. 1 is produced otherwise than 
by either equal rigidities or equal densities, the ratio of reflected 
light to incident light would not be exactly equal to (73). This 
dynamical truth was referred to in my introductory Lecture 
(pp. 15, 16, above); and I had then come to the conclusion from 
Professor Rood's photometric experiments that the observed 
amount of reflected light from glass agrees too closely with (73) 
to allow any deviation from either equal rigidities or equal 
densities, sufficient to materially improve Green's dynamical 
theory of the polarization of light by reflection. This conclusion 
is on the whole confirmed by Rayleigh's very searching investi- 
gation of the reflection of light from glasses of different kinds* ; 
but the great differences of reflectivity which he found in the 
surface of the same piece of glass in different states of polish, 
rendered it impossible to get thoroughly satisfactory results in 
respect to agreement with theory ; as we see by the following 
statement which he gives as a summing-up of his investigation. 

" Altogether the evidence favours the conclusion that recently 
" polished glass surfaces have a reflecting power differing not more 
" than 1 or 2 per cent, from that given by Fresnel's formula ; but 

* " On the Intensity of Light Reflected from certain surfaces at nearly Perpen- 
dicular Incidence," Proc. R. S., XLL pp. 275294, 1886; and Scientific Paper$, n. 
pp. 522542. 

24-2 



372 LECTURE XVIII. 

Molar. " that after some months or years the reflection may fall off from 
" 10 to 30 per cent., and that without any apparent tarnish. 

" The question as to the cause of the falling off, I am not in 
" a position to answer satisfactorily. Anything like a disintegration 
" of the surface might be expected to reveal itself on close in- 
"spection, but nothing of this kind could be detected. A super- 
" ficial layer of lower index, formed under atmospheric influence, 
"even though no thicker than 1/100000 inch, would explain a 
" diminished reflection. Possibly a combined examination of the 
" lights reflected and transmitted by glass surfaces in various 
" conditions would lead to a better understanding of the matter. 
"If the superficial film act by diffusion or absorption, the trans- 
" mitted light may be expected to fall off. On the other hand, 
" the mere interposition of a transparent layer of intermediate 
" index would entail as great an increase in the transmitted as 
"falling off in the reflected light. There is evidently room here 
" for much further investigation, but I must content myself with 
" making these suggestions." 

137. Consider next grazing incidence (i = 90, 1 = 0, a = u~ } ) 
of an equivoluminal wave of vibrations in the plane of incidence, 
and therefore very nearly perpendicular to the reflecting surface. 
We see immediately that equations (55) are satisfied by 
H = 0, H' = 0, ,H = 0, ,G = 0, G-G' = 0. 
This shows that, whether for equal or unequal rigidities, we have 
approximately total reflection, and that the phase of the reflected 
light corresponds to G' = G. Looking now to 136(71), we see 
that in the case of equal rigidities of the two mediums G'/G is 
negative for direct incidence, while we now find it to be positive 
for grazing incidence. Hence if it is real for all incidences it 
must be zero for one particular incidence. This is fundamental 
in the dynamics of polarization by reflection and of principal 
incidences. 

138. On the other hand, for equal densities and unequal 
rigidities, look to 135 (69); and we see that G'/G is positive for 
direct incidence: and by 137 it is + 1 for grazing incidence. 
Hence, it cannot vanish just once or any odd number of times 
when i is increased from to 90; but it can vanish twice or 
an even number of times. This is essentially concerned in the 



OPACITY AND REFLECTIVITY OF METALS. 373 

explanation of the remarkable discovery of Lorenz and Rayleigh, Molar. 
referred to in 81 above. 

138'. Lastly; going back to 123, we see that our notation 
has secured that, for direct incidence, C'/C is negative or positive 
just as is G'/G, according as the rigidities or the densities of the 
two mediums are equal. But at grazing incidence, C'/C, while 
still negative for equal rigidities, is positive for equal densities. 

139. So far everything before us, dynamical and experi- 
mental, confirms Green's original assumption of equal rigidities 
and different densities to account for light reflected from and 
transmitted through transparent bodies. Before going on in 
Lecture XIX. to the promised reconciliation between Fresnel and 
dynamics for transparent substances, let us, while keeping to the 
supposition of equal rigidity of ether throughout vacant space and 
throughout space occupied by ponderables of any kind, briefly 
consider what suppositions we must make in respect to our 
solution of 132 [(59) (63)], to explain known truths regarding 
the reflection of light from metals or other opaque bodies. 

140. The extremely high degree of opacity presented by all 
metals for light of all periods, from something considerably longer 
than that of the extreme red of the visible spectrum (2'5 . 1Q-" sees, 
for A line) to something considerably less than that of the extreme 
violet (1-3 .10-" sees, for H line), is the most definite of the visible 
characteristics of metals : while the great brilliance of light re- 
flected from them, either directly or at any angle other than 
grazing incidence*, compared with that reflected from glass or 

* As obliquity of incident light is increased to approach more and more nearly 
grazing incidence, the brilliance of the reflected light approaches more and more 
nearly to equality with the incident light. At infinitely nearly grazing incidence 
we find theoretically ( 137) total reflection from every polished surface; polish 
being defined as in 86 above. Indeed we find observationally iu surfaces such as 
sooted glass, which could scarcely be called polished according to any definition, a 
manifest tendency towards total reflection when the angle of incidence is increased 
to nearly 90. 

A striking illustrative experiment may be made by placing, on a table covered 
with a black cotton-velvet tablecloth, two pieces of plate glass side by side, with an 
arrangement of light and screen as indicated in the accompanying sketch. L is a 
lamp which may be held by hand, and raised or lowered slightly at pleasure. OO' 
is an opaque screen. PP 1 is a screen of white paper resting on the table. It would 
be startling, if we did not expect the result, to see how much light is reflected from 



374 



LECTURE XVIII. 



Molar, crystals or from the most brilliantly polished of commonly known 
and seen non-metallic bodies, is their most obvious arid best known 
quality. The opacity of thin metal plates hitherto tested for all 
visible lights from red to violet has been found seemingly perfect 
for all thicknesses exceeding 3.10~ 5 cm. (or half the wave-length 
of yellow light in air). "When, in the process of gold- beating, 
the thickness of the gold-leaf is reduced to about 2.1()- 5 cm. (or 
about one-third of the wave-length of yellow light) it begins to 
be perceptibly translucent, transmitting faint green light when 
illuminated by strong white light on one side. The thinnest of 
ordinary gold-leaf (7 .10~ 5 cm., or about one-eighth of the wave- 
length of yellow light) is quite startlingly translucent, giving a 
strong green tinge to the transmitted light. Silver foil l'5.10~ 5 cm. 
thick (considerably thinner than translucent gold-leaf) is quite 
opaque to the electric light so far as our eyes allow us to judge ; 

the soot above the boundary of the shadow of 00'. The experiment is rendered 
still more striking by placing a flat plate of polished silver beside the two glass 
plates, and seeing how nearly both the sooted and the clean glass plates come in 
rivalry with the silver plate in respect to totality of reflection, when L is lowered to 



* 



more and more nearly grazing incidence of its light. It is interesting also to take 
away the paper screen, and view the three plates and the lamp by an eye placed in 
positions to receive reflected light from the three mirrors, sooted, polished glass, 
and polished silver. 

Another interesting experiment may be made by looking vertically downwards 
through a Nicol at the three surfaces, or at a clean surface of mercury or water, 
illuminated by light from L at nearly grazing incidence in a dark room. A sur- 
prisingly large amount of light is seen from the sooted surface, and is found 
to be almost wholly polarized to vibrations perpendicular to the plane through L 
and the line of vision. If the glass and silver are very well polished and clean, 
little or no light will be seen from them ; unless L is very intense, when probably 
a faint blue light, polarized just as is the light from the soot, will be visible, 
indicating want of molecular perfectness in the polish, or a want of optical 
perfectness in the most perfect polish possible for the molecular constitution of 
the solid or liquid, according to the principles indicated in 86 above. 



ELECTROMAGNETIC SCREENING AND TRANSPARENCY. 375 

but it is transparent to an invisible violet light through a small Molar, 
range of wave-length from about 3-07.10~ s to 3'32.10- 5 cm.* 
(periods from 1-02. lO' 15 to Ml. 10-" of a second). 

141. The extreme opacity of metals is quite lost for Roentgen 
rays (which are probably light of much shorter period than 10~ 1S of 
a second); sheet aluminium of thicknesses up to two or three 
centimetres being transparent for them. For some qualities of 
Roentgen rays even thick sheet lead is not perfectly opaque. 

142. We have no experimental knowledge in respect to the 
opacity of exceedingly thin metallic films for radiant heat of 
longer periods than that of the reddest visible light. It seems 
not improbable that through the whole range of periods up to 
2.10~ 1S of a second, through which experiments on the refractivity 
and reflectivity of rock-salt and sylvin have been made by Langley, 
Rubens, Paschen, Rubens and Nicols, and Rubens and Aschkinaas 
(see above, Lecture XII. p. loO), the opacity of gold-leaf and other 
of the thinnest metal foils may be as complete, or nearly as com- 
plete, as it is for visible light. 

143. But, when we go to very much longer periods, we 
certainly find these thin metal foils quite transparent for variations 
of magnetic force. Thus, sheet copper of thickness 7.10~ 8 cm. 
(about a wave-length of orange light in air), though almost per- 
fectly opaque to variation of magnetic force of period one-third or 
one-fourth of 1/(8.10 9 ) of a second, is almost perfectly transparent 
for periods of magnetic force of three or four times 1/(8.10 9 ) of 
a second, and for all longer periods. 

Taking greater thicknesses, we find a copper plate two milli- 
metres thick, almost a perfect screen, that is, almost perfectly 
opaque, in respect to the transmission of the magnetic influence 
of a little bar-magnet rotating 8000 times per second f; somewhat 
opaque, but not wholly so, when the "speed is 100 times per second; 
almost, but not perfectly, transparent, that is to say, very slightly 

* See pp. 185, 186 of Popular Lectures and Addresses, Vol. i., Ed. 1891 (Friday 
Evening Royal Institution Lecture, Feb. 3, 1883), where experiments illustrating the 
reflectivity and the transparency of some exceedingly thin films of platinum, gold, 
and silver, supplied to me through the kindness of Professor Dewar, are described. 

t See 3 of Appendix K, "Variational Electric and Magnetic Screening," 
reprinted from Proc. R. S. Vol. XLIX. April 9, 1891 ; also Math, and Phys. Papers, 
Vol. in. Art. en. "Ether, Electricity, and Ponderable Matter," 35. 



376 LECTUUK XV11I. 

Molar, if perceptibly obstructive, when the speed is once per second ; not 
perceptibly obstructive, when the period is 10 seconds or more. 
Now, according to what is without doubt really valid in the 
so-called electro-magnetic theory of light, we may regard as a 
lamp, a bar-magnet rotating about an axis perpendicular to its 
length, or having one pole caused to vibrate to and fro in a straight 
line. We may regard it as a lamp emitting light of period equal 
to the period of the rotation or of the vibration. For the light 
of this lamp, sheet copper two millimetres thick is almost per- 
fectly transparent if the period is anything longer than one second ; 
but it is almost perfectly opaque if the period is anything less than 
1/8000 of a second down to one eight hundred million millionth of 
a second (the period of extreme violet light); and is probably quite 
opaque for still smaller periods down to those of the Roentgen 
rays, if we regard these rays as due to vibrators giving after each 
shock a sufficient number of subsiding vibrations to allow a period 
to be reckoned. Whatever the distinctive characteristic of the 
Roentgen light, sheet copper two millimetres thick is perceptibly 
translucent to it, and sheet aluminium much more so. 

144. We may reasonably look for a detailed and satisfactory 
investigation, mathematical and experimental intelligence acting 
together, by which we shall thoroughly understand the continuous 
relation between the reflection and translucence of metals and 
transparent bodies, and the phenomena of electric and magnetic 
vibrations in insulating matter, in non-magnetic metals, in soft 
iron, and in hardened steel, for all vibrational periods from those of 
the Roentgen rays to ten or twenty seconds or more. The in- 
vestigation must, of course, include non-periodic motions of ether 
and atoms. It cannot but show the relation between the electric 
conductivity of metals and their opacity. It must involve the 
consideration of molecular and atomic structures. Maxwell's 
electro-magnetic theory of light was essentially molar*; and there- 
fore not in touch with the dynamics of dispersion essentially 

* " Suppose, however, that we leap over this difficulty [regarding electrolysis] by 
"simply asserting the fact of the constant value of the molecular charge, and that 
"we call this constant molecular charge, for convenience in description, one 
"molecule of electricity. 

"This phrase, gross as it is, and out of harmony with the rest of this treatise, 
"will enable us at least to state clearly what is known about electrolysis, and to 
"appreciate the outstanding difficulties." Maxwell, Electricity and Magnetism, 
Vol. i. p. 312. 



GREEN'S DYNAMICS OF REFLECTION AND REFRACTION. 377 

involved in metallic reflection and trauslucency ; though, outside Molar. 
his electro-magnetic theory, he was himself one of the foremost 
leading molecularists of the nineteenth century: witness his 
kinetic theory of gases; and his estimates of the sizes and 
weights of atoms ; and his anticipation of the Sellmeier-Helmholtz 
molecular dynamics of ordinary and anomalous dispersion, in a 
published* Cambridge Examination question. 

145. First, however, without any molecular hypothesis, and 
without going beyond Green's purely molar theory of infinite 
resistance to compression, and equal effective rigidities of ether in 
all bodies and in space void of ponderable matter, let us try how 
nearly we can explain the high reflectivity and the great opacity 
of metals. Either great rigidity, or great density, or both great 
rigidity and great density, of ether in metals would explain these 
two properties : but we have agreed not to assume differences of 
rigidity, and there remains only great density. It is interesting 
to remark however, that infinite rigidity would give exactly the 
same law as infinite density ; because each extreme hypothesis 
would simply keep the ether at the interface absolutely unmoved ; 
and this even if we allow the ether to be compressible within 
liquids and solids, as we are going to do later on. 

146. Go back now to 132 (59) (63), and, following 
Green, make v = oc , and ,v = ao . This makes, by (50), c = ,c = <uf; 
and reduces (61) and (59) to 

L = Kat ........................... (74), 

and 



, .. 

where K > ' p) ' , <#> - tan- -* * = tan- -* 



......... (75). 

* Rayleigh, in a footnote appended in 1899 to the end of his paper "On the 
Reflexion and Refraction of Light by Intensely Opaque Matter 1 ' (/'/<//. Mag. 1872; 
republished as Art. xvi. of his Scientific Papers, Vol. i.), writes as follows: 

"I have lately discovered that Maxwell had (earlier than Sellmeier) considered 
the problem of anomalous dispersion. His results were given in the Mathematical 
"Tripos Examination, Jan. 21, 1869 (see Cambridge Calendar for that year)." 

In this examination question the vitcotu term, subsequently given by Helmholtz, 
is included. 

t Take -at (not +ac, a being positive, in order that, in 128, //, of the upper 
medium may have extinctional factor e"*** 1 ', and, H, of the lower, may have 
extinctional factor "*. See footnote on 158". See also 128 and (49), and (54). 



378 LECTURE XVIII. 

Molar. This agrees with Green's result*; and the square of the first 

ni 

factor of the final expression for -~ is the formula by which 

curve 3 of figs. 4, 5, 6, for water, flint-glass, and diamond, in 
102, 103 above, were calculated. 

147. To realize this solution in the most convenient way for 
physical optics, put 

/(#) = e 1 " 9 = cos o)0 + t, sin wQ (76) ; 

and take 

for incident wave 6 = t- (77), 

with same continued for reflected wave, where s denotes a space 
in the path of the incident ray, continued in the reflected ray. 

Use these in 128, and for the real problem take the real part 
of each expression so found. We thus have, for the vibrational 
displacements, 



Incident wave, G cos &> [ t 

V u 

Reflected wave, 



...(78). 



148. Let now for a moment ,p/p = x . This makes 
Kj,p = 1 ; <f> = >/r = i ; and gives, for vibrational displacement in 
the reflected wave 

t-- 



L (t 

[_ v 



Thus the formulas show, for vibrations in the plane of inci- 
dence, that, at every incidence, the reflection is total (which we 
know without the mathematical investigation, because there is 
no loss of energy) ; and that the reflected ray is advanced in phase 
TT 2i relatively to the incident ray. The former proposition is 
proved for vibrations perpendicular to the plane of incidence by 
the formula (39) of 123 ; but with just TT for phasal change. 
Hence incidence at 45 instead of the 70 to 76 of metals ( 99 
above), gives 90 of phasal difference ; and it is easy to see from 

* Green, Mathematical Papers, p. 267. 



QUASI WAVK-MnTln\ WITH XEUATIVK OR IMAGINARY DENSITY. 879 

the formulas that values of ,p/p large enough to give the Molar. 
brilliance of metallic reflections could not give any approach 
to the elliptic and circular polarizations which observations show 
( 99 above) in all metallic reflections. Thus, though our trial 
hypothesis of great effective density of the ether in the substance 
could give reflections, and therefore general appearances, un- 
ili.-tinguishable to the naked eye from what we see in real metals, 
it fails utterly to explain the qualities of the reflected light 
discovered by polarizational analysis. Seeing thus that no real 
positive effective density in the substance can explain the qualities 
of metallic reflection, we infer that the effective density is negative 
or imaginary ; and thus we are led by strictly dynamical reasoning 
to the brilliant prevision of MacCullagh and Cauchy that metallic 
reflection is to be explained by an imaginary refractive index. 

149. In 159 below we shall find, by a new molecular 
theory to which I had been led by consideration of very different 
subjects, a perfectly clear and simple dynamical explanation of a 
real negative quantity for effective density of ether traversed 
by light-waves of any period within certain definite limits, in a 
space occupied by a solid or liquid or gas. We shall also find 
definite molecular and dimensional conditions which may possibly 
give us a sure molecular foundation for an imaginary effective 
density; though we are still very far from a thorough working- 
out of the full dynamical theory. 

We conclude the present Lecture with a short survey of the 
quasi wave-motion which can exist in an elastic medium having 
a definite negative or imaginary effective density ,p ; and of the 
reflection of light at a plane surface of such a medium. 

150. Let the rigidity be w, real ; and the density ,p, an 
unrestricted complex as follows 

f p = nr (cos< i sin^>) ..................... (79), 

where v and <f> are real, and for simplicity may be taken both 
positive. This gives for propagational velocity of equivoluminal 
waves 

,u = w (cos </> + i sin <); ,u~ l = w~ l (cos ^ - 1 sin </>U 



In 
where w = * / , a real velocity 



380 LECTURE XVIII. 

Molar. For displacement at time t, considering only equrvoluminal wave- 
motion, let rj -f it{ denote displacement in a wave-plane perpen- 
dicular to OX ; and choosing for our arbitrary function 



we have 77 + irj' = 6 *(t-*M ..................... (81); 

whence, taking ,u~ l from (80), we find, as a real solution, 

^= 6 -8mJ**/'COSG>(--COS^#/W) ............ (82). 

This shows that the wave-plane travels in the + ^--direction with 
velocity w sec ^<f>, and with vibrational amplitude diminishing 
according to the exponential law - !**/ It is interesting to 
remark that the propagational velocity of this subsidential wave is, 
in virtue of the factor seci<, essentially greater than the velocity 
in a medium of real density equal to the modulus, vr, of our 
imaginary density; and is infinite when < = 90. This illustrates 
Quincke's discovery of greater velocity of light through a thin 
metallic film than through air. 

| 151. Remark now that, according as the real part of the 
complex, ,p, is positive or negative, < is <90, or >90 ; that, in the 
extreme case of ,p real positive, </> is zero ; and that, in the other 
extreme case (,p a real negative quantity), <f> is 180. In every 
case between those extremes, ^<f> is between and 90, and 
therefore both cos ^ and sin A(/> arc positive. This implies loss 
of energy in the inward travelling wave : except in the second 
extreme case, ^</> = 90, when its propagational velocity is infinite ; 
and (82) becomes 

77 = -*/> sin cot ........................ (83), 

which represents standing vibrations of ether in the substance, 
diminishing inwards according to the exponential law ; and 
therefore proves no continued expenditure of energy. We con- 
clude that, for our ideal silver, the effective quasi-density of 
ether in the metal is essentially real negative', but that for all 
metals of less than perfect reflectivity it must be a complex, of 
which the real part may be either negative or positive. 

152. Confining our attention for the present to ideal silver, 
and to merely molar results of the molecular theory promised for 
Lecture XIX., let us put, in 123, 132, 

-/*'=* * a ............................ (84), 



INFINITE VELOCITY DISPROVED FOR CONDENSATIONAL WAVES. 381 

where i>* denotes a real positive numeric ; and let us find the Molar 
difference of phase between the two components, given respec- 
tively by 123 and 132, due to incident light polarized in 
any plane oblique to the plane of incidence. That difference of 
phase is 90 for Principal Incidence (see 97 above). I have 
gone through the work with Green's supposition v = oo , ,v = oo , 
in 132 ; and using the formula for G'/G so given in 147, I have 
found that for all values of i/ 5 from to oo it makes the Principal 
Incidence between and 45. Now observation gives the 
Principal Incidences for all colours of light and all metals, 
between 45 and 90 (see table of 99, showing for all cases 
of metallic reflection hitherto made, so far as generally known, 
Principal Incidences ranging from 66 to 78). Hence polarizational 
analysis of the reflected light as thoroughly disproves, for metallic 
reflection, Green's assumption of propagational velocity infinite 
for condensational -rare factional waves within the metal, as it was 
disproved for transparent substances in 104, 105 by mere 
observation of unanalysed reflectivities. 

153. Hence, anticipating Lecture XIX. as we did in 133 
above, let us, while still keeping v = x in the ether outside the 
metal, now make ,v small enough inside the metal to practically 
annul L. This reduces (39) and (59) of 123 and 132, to 



_C" ,6-6. -Q> lp b-p,b 

-~ = -~ = 






Putting now in (39) and in this 



= r; ^ = -*; - = - tv * (86), 

b ,u - 



p 



, COS .1 

where 



cos v cos 



C' ivrl vr i 

* efind -e--r^+i -.-+; (88) ' 

and I__ =: ^l (89). 

G v + ir 

* The sign minus is chosen here in order that (** not -w may be the reducing 
factor of ,O ; p- being a positive length. See foot-notes on 146, 158". 



382 

Molar. Put now 



LECTURE XVIII. 



tan' 1 = 0, 
vr 



tan" 



This reduces (88) and (89) to 



C 
-- 
U 



and 



cos0 tsm0 . _ 

= - : ^ = cos 20 - i sin 20 
cos + t sm 

cos ^ i sin 



cos ^ + i sin ^ 



cos 2^ - i sin 2 



.(92). 



These formulas express total reflection for the two cases re- 
spectively of vibrations perpendicular to the plane of incidence, 
and vibrations in this plane ; and they give us 20 and 2^, which 
we may, for brevity, call " the phases of the vibrations." Thus, 
calling P the plane of incidence and reflection, we find 

Phase of vibration perpendicular to P phase of vibration in P 
= 2(^-0) (93) 

This means that the vibration perpendicular to P precedes the 
other by 2 (* - 0). 

Whatever positive value v" has, this difference is essentially 
zero for i=0; and \ve find that it increases through +90 to 
+ 180 when i is increased from to 90; as illustrated in the 



accompanying tables for the two cases, v- 



10. 



, 1 


r 




i = tun 


^ tftn - 


2 (^ 0) 




vr 


V 







547 


547 


o-o 


10 


53-5 


55-9 


4-8 


20 


50-1 


59-1 


18-0 


30 


45-0 


63-4 


36-8 


40 


38-7 


68-2 


59-0 


50 


31-7 


72-9 


82-4 


53-2 


29-3 


74-3 


90-0 


60 


24-1 


77-4 


106-6 


70 


16-2 


81-7 


1310 


80 


8-2 


85-9 


155-4 


90 


o-o 


90-0 


180-0 



PRINCIPAL INCIDENCE FOR IDEAL SILVER. 



383 



1 


<=<": 


^ = tan- r 

r 


2(*-0) 





175 


175 


o-o 


10 


17-3 17 H 


10 


20 


16-5 18-7 


4-4 


30 


15-1 20-3 


10-4 


40 


13-4 22-8 


18-8 


50 


114 


W-fl 


31-2 


60 


8-7 


33-3 


49-2 


70 


5-9 


440 


76-2 


738 


4-8 


49-7 


89-8 


80 


3-0 


62-3 


118-6 


90 


<M> 


900 


180-0 



Molar. 



I/* =10. 

154. The angle of incidence (53"'2 for v* = '5, 73'8 for 
i? = 10) which gives the 90 advance of phase is what has been 
defined as the Principal Incidence ( 97 above). The reflectivities 
for the two polarizational components of incident light being 
perfect, the azimuth which, for the "Principal Incidence," gives 
circular polarization for the reflected light, that is, the " Principal 
Azimuth," is 45. For all real metals observation shows the phase- 
difference 2 (> - 6} to be positive, that is to say, the vibration 
in the plane of reflection to lag behind the vibration perpendicular 
to it ; as we find it for ideal silver by dynamical theory. 

155. The easiest mathematical method for finding the 
Principal Incidence, /, for any given value of i/ 8 is algebraic, as 
follows. For Principal Incidence we have 

2(3-0) = 90 ............... . ........... (94). 

This gives tan 2^ tan 20 = - 1, or in algebra, from (90), 
40%* 



= 






whence, 

^-^(^1 + 4 + -) + 1 = .................. (96). 

Taking the greater root of this regarded as a quadratic for r*, 
and substituting it for r* in the general expression 

tan' = ^(r -l)/(i/ +1) ......... . ...... ....(97) 

given by (87) for t in terms of v and r, we find tan 5 / for the 



384 



LECTURE XVIII. 



Molar. Principal Incidence. (The less root of the quadratic is rejected, 
because it makes tan-i negative.) 

The following table has been thus calculated directly, to show 
for fourteen values of v or v'-, the Principal Incidence, /. 



V 


V* 


tan 7 


' 


o-oo 


o-oo 


1-000 


45-0 


0-50 


0-25 1-193 


50-0 


0-71 


0-50 1-333 


53-1 


1-00 


1-00 1-554 


57-2 


1-41 


2-00 1-887 


62-1 


2-00 


4-00 


2-387 


67-2 


2-45 


&-00 


2-786 


70-3 


3-16 


10-00 


3-438 


73-8 


3-74 


14-00 


3-982 


75-9 


4-00 


16-00 


4-223 76-6 


4-47 


20-0 


4-680 77-9 


G-OO 


36-0 


6-160 80-8 


10-00 


100-0 


10-10 84-3 


14-00 196-0 


14-07 85-9 


20-00 


400-0 


20-05 87-1 



156. The converse problem of finding i/ 2 for any given 
Principal Incidence, by (96) and (97), yields a cubic equation 
for i/ 2 . The table of 155 proves that this cubic equation has 
one, and only one, real positive root for every value of / between 
45 and 90 ; and no real positive root for values of / between 
and 45. For our present purpose it is most easily solved by 
trial and error, aided by the table of 155. I have thus found for 
the three Principal Incidences measured by Conroy for red, 
yellow, and blue light respectively, incident on his silver film 
polished with putty powder (table of 99 above), the following 
values of v and v~. 





I 


v v- 


Red 


76 29' 


3-9 15-2 


Yellow 
Blue 


74 37' 
71 33' 


3-3 10-9 

2-7 7-29 



157. The diagram of 88 shows that Conroy 's silver film, 
polished as he polished it with putty powder, may be regarded as 
almost our ideal silver : this view is confirmed by his three 



"ADAMANTINISM" AND UNEQUAL RIGIDITIES IN METALS? 385 

Principal Azimuths, 43 51', 4352', 43 0', being each as nearly Molar. 
a good approximation to 45 as it is. (See 159" below.) But 
their shortcomings of from one to two degrees below 45 are no 
doubt real, and point to the correction of the real values of v- 
by the addition of small purely imaginary terms. Thus, to fit 
the formulas of 150 to Conroy's silver, we may, keeping tor a 
positive quantity, take </> = 180 ^; where ^, which would be 
zero for ideal silver, may for real silver, have some small value of 
a few degrees. 

158. It would be interesting to pursue the subject further, 
and include with silver, other metals for which we have the 
experimental data such as those shown in the table of 99. 
To do this, we may conveniently in (87), (88), (89), put 

v* = p(cosx-isiu x ) (98); 

and use the experimental data of Principal Incidence and 
Principal Azimuth to determine in each case the two unknown 
quantities p cos % and p sin ^ : but time forbids. This would 
be, in fact, a working out of the theory, or empirical formula, of 
MacCullagh and Cauchy, to comparison with observational results 
regarding metallic reflection, such as has been done by Eisenlohr, 
Jamin, Conroy, and others*. The agreement of the theory with 
observation has been found somewhat approximately, but not 
wholly, satisfactory. Stokes, as communicator of Conroy's paper, 
No. in. (Proc. R.S., Feb. 15, 1883), comments -on this want of 
perfect agreement; and suggests that it is to be accounted for by 
the inclusion in respect to metallic reflection of what he proposes to 
call "the adamantine property" of a substance ; being the property 
required to explain the deviation, from Fresnel's law of reflection 
of light by transparent bodies, discovered more than eighty years 
ago by Airy, in diamond. This adamantine property, as we 
shall see in Lecture XIX., 173, is to be explained dynamically 
by assuming a small imaginary iq, for the velocity of condensa- 
tional- rarefactional waves in the substance ; not small enough to 
utterly annul L in our formulas of 132. Its magnitude is 
measured by the ratio q/u, which I propose to call adamantinism. 
Some deviation from exact equality between the effective rigidities 
of ether in the two mediums might also be invoked to aid in 
procuring agreement between dynamical theory and observation, 

* Basset, Physical Optics, 371, 380; Mascart, Traitf d'Optique, Vol. u. 
T. L. 25 



386 LECTURE XVIII. 

Molar, for reflection of light from all substances, transparent, or metallic 
or otherwise opaque. 

158'. Meantime, I finish this Lecture XVIII. on the Re- 
flection of Light with an application of 123 (39), and 132 (59) 
with L = 0, to the theory of Fresnel's rhomb. 

Let the medium in which u is the velocity of light be a trans- 
parent solid or liquid ; and let the other medium be undisturbed 
ether. We cannot now continue Green's convenient usage adopted 
in 116, and call the former "the upper" and the latter "the lower." 
We now have 
>?<; p>,p', ,i>i\ ,u]u=n\ /a/,p=fi a ; sin ,i = /* sin i ; b=cosi/u; ,&=cos,t'/,w... 

For convenience, to suit the case of ,i > i, write as follows, 
the equations cited above, 

V _6 6> G'_ P ,b-,pb 
C~b + ~,b' G~p,b + ,pb" 

the sign minus being transferred from their first members because, 
in virtue of ,i > i, C'fC is now positive through the whole range of 
non-total reflection, [t = to t = sin -1 (I//*) giving ,t = 90]; and 
G'/G is positive through the range up to the Brewsterian angle of 
zero reflected ray [i = to i = tan" 1 (l//^)]. From i = tan" 1 (l//t) to 
i = sin"" 1 (1 //it), G'/G decreases from to 1; and it is imaginary- 
complex of modulus 1, through the whole range of total reflection 
[t^sin-^l//*) to i = 90 e ]. 

158". When i is between sin -1 (l//i) and 90, ,6 2 is real 
negative ; and, denoting its value by q 2 (as in 118), we have 

, */(/** sin a il) r>///\ 

b = -LQ*\ where q = b v ^ . ' ......... (98"). 

H cos i 

Remark that q increases from to oo when i is increased from 
sin- 1 (l/fi) to 90. Using (98'") in (98") we find, 



whence, by De Moivre's theorem, 



-^ =cos 2i/r + i sin 2-^ ; -^ =cos2^; tsin 2^ (98 V ), 

where -/r = tan- 1 ^; % = tan" 1 - (98 vi ). 

* The sign mhnts is here chosen in order that fiw, not e'V^v, may be the 
reducing factor of ,C and ,G for the disturbance in the ether outside, see 128; 
y being zero at the reflecting surface, and negative outside. 



PHASAL CHANGES BY INTERNAL REFLECTION. 



387 



By (98 Ti ) and (98"') we see that i/r increases from to 90 and Molar. 
decreases from 90 to 0, when t is increased from 
sin- 1 (/i- 1 ) to 90. 

158'". Putting for a moment q/b = x, we find 




Equating this to zero we find, r + x) a mnmum, 

qjb = p-\ which makes ^ = X = tan~> (/*->) ...... (98'); 

and to make q/b = /*-', we have tan't = 2/(/i - 1). We infer that 
when i is increased from sin" 1 (/A -I ) to tan" 1 v r [2/(/* 8 1)], (^ + x) 
decreases from 90 to 2tan~ 1 (/A~ 1 ) and then increases to 90 
again, when t is increased to 90. This will be useful to us 
in 158*. 

158". Realising now, as in 147, we find real solutions as 
follows, for vibrations perpendicular to, and in, the plane (P) of 
incidence and reflection : 



....(98"); 



...(98'). 



Vibrations 
perpendi- -I 
cular to P I Reflected wave C cos 



Vibrations 
inP 



Incident wave C cos o> ( t 

V w 

[(-*)+ 

Incident wave G cos CD ( t ) 

\ uJ 



Reflected wave G cos a> (t - -) - 2^ 



Thus we see that the reflected vibrations in P are set back in 
phase by 2 (^ + ^) relatively to the reflected vibrations perpendi- 
cular to P. By 158'" we see that this back-set is 180 at the 
two incidences, t = sin" 1 (/i" 1 ), and t = 90 (the limits of total 
reflection); and that it decreases to a minimum, 4 tan" 1 ^" 1 ), at 
the intermediate angle of incidence i = tan" 1 V[2/(/* 2 1)]. 

The back-sets of the phase through the whole range of total 
reflection for four refractive indices, 1'46, 1'5, 1 - 51, and T6 are 
shown in the Table II. of 158 T below. 

Column 1 represents angles of incidence. 

252 



388 



LECTURE XVIII. 



Molar. Columns 2 and 3 of Table II. represent the absolute back-sets 
of phase by reflection respectively in, and perpendicular to, P. 

Column 4 represents the back-set of phase of vibrations in P, 
relatively to vibrations perpendicular to P, produced by a single 
reflection. 

The numbers in Column 4, doubled, are the total back-sets by 
two reflections, at the same angle of incidence i, of vibrations in, 
relatively to vibrations perpendicular to, P. Each of these numbers 
is between 180 and 360; and therefore it indicates virtually a 
setting- for ward of the phase of vibrations in P, relatively to the 
phase of those perpendicular to P, by the amount to which each 
number falls short of 360. Hence the numbers in Column 5 of 
Table II. represent virtual advances of phase of the vibrations in, 
relatively to the vibrations perpendicular to, P, after two reflections. 

158 T . It will be seen that all the phasal differences shown 
in Column 4 of Table II. are obtuse angles : with minimum values 
shown in Table I., which is extended to include zinc sulphide, 



TOTAL INTERNAL REFLECTION. TABLE I. 



Refractive index 
H 


Brcwsterian 
angle 
tan-' (l//x) 


Limiting inci- 
dence for total 
internal 
reflection 
sin-i (L 


Incidence for 
minimum phasal 
difference 
tan-\/[2/( M -l)] 


Minimum 
phasal 
difference 
4 tan-i (1/ M ) 


1-46 


34 24'-5 


43 14' 


53 3' 


137 38' 


1-50 


33 41-25 


41 49 


51 40 


134 45 


1-51 


33 30-75 


41 28 


51 20 


134 3 


1-60 


32 0-25 


38 41 


48 33 


128 1 


(Zinc Sulphide) 2-371 


22 52 


24 57 


33 20 


91 28 


u. = cot 22 = 2-414 


22 30 


24 28 


32 4G 


90 


(Diamond) 2-434 


22 20 


24 15 


32 31 


89 20 


(Realgar) 2-454 


22 10 


24 3 


32 15 


88 40 



diamond, and realgar. By Table I. we see that the phasal differ- 
ences for internal reflection in glasses, and all known transparent 
bodies of refractive index less than 2 '4 14, are obtuse for all angles 
of incidence through the whole range of total internal reflection. 
This conclusion was very startling to myself, because for eighty 
years we have been taught that, for total internal reflection in 
glass, the phasal difference was an acute angle in a single re- 
flection ; and that it was 45 for each reflection in the Fresnel 



REFLECTION OF CIRCULARLY POLARIZED LIGHT. 389 

rhomb, instead of 135 which we now know it is. How it is so, Molar, 
we can easily see by the following simple considerations. 

(1) Circularly polarized light in every case of direct incidence 
and reflection (metallic or vitreous, external or internal), has its 
circular orbital motions in the same absolute directions in the 
direct and the reflected waves. Hence if the orbital motion ideally 
seen by an eye receiving the incident light is anti-clockwise, it is 
clockwise to an eye receiving the reflected light. (Hence the 
arrangement in respect to signs in figs. 3 a and 3 b of 92 above 
is explained thus: First imagine the two diagrams as corre- 
sponding to normal incidence: for the incident light, fig. 36, with 
its face down, looked up to; and for the reflected light, fig. 3a, 
with its face up, looked down on. Then incline the two planes 
continuously to suit all incidences from i = 0, to i = 90.) 

(2) Taking first, external reflection of light, of circular anti- 
clockwise orbits, incident on glass of negligible adamantinism * ; 
increase the angle of incidence from to the Brewsterian tan" 1 /*. 
The motion constituting the reflected light is in clockwise elliptic 
orbits, of increasing ellipticity, with long axes perpendicular to 
the plane of reflection, till at tan" 1 /* it is rectilineal. Increase 
now the angle of incidence to 90 : the orbital motions are elliptic 
anti-clockwise with diminishing ellipticity (long axes still perpen- 
dicular to the plane of reflection) ; and become infinitely nearly 
circular when the incidence is infinitely nearly grazing. 

(3) Proceed now as in (2), but with internal instead of 
external reflection at a glass surface: the incident circularly 
polarized light being anti-clockwise. With increasing incidences 
from t=0, the reflected light is clockwise elliptic with increasing 
ellipticity, till it becomes rectilineal at the Brewsterian tan-^l//*). 
Increase i farther : the reflected light is now anti- clockwise elliptic, 
with diminishing ellipticity till it becomes circular at the limit of 
total internal reflection, suv^l/^). Increase i farther up to 90: 
the reflected light is elliptic, still anti-clockwise, with ellipticity 
increasing to a maximum when % = sin' 1 V[2/(/*- 1)] and 
diminishing till the orbit becomes again infinitely nearly circular 
when i is infinitely little less than 90. 

* For the change from the present statement, required by adamantinism when 
perceptible, see below, 178182. 



390 LECTURE XVIII. 

Molar. (4) With, instead of glass, an ideal substance whose refrac- 
tive index is 2-414; anti-clockwise circularly polarized light, 
incident at angles increasing from 24 28', gives total reflection, 
and anti-clockwise circular orbits, becoming elliptic, with ellip- 
ticities increasing to rectilineal vibrations when i = 32 46'. 
Farther increase of i up to 90 gives elliptic orbits of ellipticities 
diminishing to circularity at i = 90: all anti-clockwise. 

(5) Diamond and realgar, and all other substances having 
H > 2'414, have three principal incidences for internal reflection; 
one with non-total reflection at the Brewsterian angle tan-'(l//i); 
and two within the range of total reflection ; one less, the other 
greater, than the incidence which makes the phasal difference 
a minimum. Anti- clockwise circularly polarized light, incident 
at any angle between the last-mentioned two principal incidences, 
gives clockwise elliptic orbits for the reflected light ; presenting 
a minimum deviation from circularity (minimum ellipticity we 
may call it) at some intermediate angle. 

(6) It is interesting to remark that in every case, according 
as the phasal difference is obtuse or acute, the reflected light is 
anti-clockwise or clockivise, when the incident light is circular 
anti-clockwise. This with 158 xvii gives a simple means for ex- 
perimentally verifying the statements (1), (2), (3), (4), (5). The 
experimental proof thus given of the truth, that the phasal 
difference in every case of approximately grazing incidence is 180, 
is very easy and clear; as it can be arranged to show simul- 
taneously to the eye the incident and the reflected light ; both 
extinguished simultaneously by the same setting of the analysing 
fresnel (or quarter- wave plate) and nicol. 

(7) The rule of signs referred to in (1) of the present section, 
and illustrated in figs. 3 a and 36 of 92, must be taken into 
account to interpret the meaning of phasal difference of two recti- 
lineal components of reflected light, due to incident rectilineally 
vibrating light. Thus we see that if the vibrational plane of the 
incident light is turned anti- clockwise from P, the orbital motions 
of the reflected light are anti-clockwise, or clockwise, according as 
the component vibrating in, is set back or advanced, relatively to 
the component perpendicular to P. See 158 xvii below. 



TOTAL INTERNAL REFLECTION. TABLE II. 



Col. 1 


Col. 2 


Col. 3 


Col. 4 


Col. 5 


I 


<fr and x calculated only 
to the nearest minute, 
except in the case of 
their equality 


2 (*+*); being actual 
phasal back-set of 
vibrations in, rela- 
tively to vibrations : 
perpendicular to, 
P, produced by one 
reflection 


virtual pna'sal mf- 
vance of vibrations 
in, relatively to vi- 
brations perpen- 
dicular to, P, pro- 
duced by two reflec- 








N 


X 




tions 


41 48'-6 


0' 


180 (X 


180 0' 


0' 


42 


8 50 


160 18 


169 6 


21 48 


43 


22 14 


132 18 


154 32 


50 56 


44 


30 22 


117 36 


147 58 


64 4 


45 


36 52 


106 14 


143 6 


73 48 


46 


42 30 


97 36 


140 6 79 48 


47 


47 34 


90 28 


138 2 83 56 


48 


52 16 


84 20 


136 36 86 48 


49 


56 40 


79 


135 40 88 40 


50 60 50 


74 16 


135 6 89 48 


51 64 48 


70 


134 48 90 24 


51 40-3 67 22-8 


67 22-8 


134 45-6 (min.) 90 28-8 (max.) 


52 c.s :<r, 


66 10 


134 46 90 28 


53 7-' H 


62 38 


134 56 90 8 


53 12 






90 


54 


75 54 


59 22 


135 16 89 28 


55 


78 4 


57 26 


135 30 


89 


56 


82 46 


53 32 


136 18 


87 24 


57 


86 6 


50 52 


136 58 


86 4 


58 


89 22 


48 24 


137 46 


84 28 


59 


92 34 


46 2 


138 36 


82 48 


60 


95 44 


43 48 


139 32 


80 56 


70 


125 22 


25 52 


151 14 


57 32 


80 


153 4 


12 8 165 12 


29 36 


90 


180 





180 





43 13-8 





180 


180 





44 


18 18 


142 6 


160 24 


39 12 


45 


27 56 


124 22 


152 18 


55 24 


46 


35 8 


112 


147 8 


65 44 


47 


41 14 


102 34 


143 48 


72 24 


48 


46 38 


94 52 


141 30 


77 


49 


51 34 


88 18 


139 52 


80 16 


50 


56 10 


82 38 


138 48 


82 24 


51 


60 32 


77 36 


138 8 


83 44 


52 


64 40 


73 6 


137 46 


84 28 


53 


68 38 


69 1 


137 39 


84 42 


53 2-98 


68 49-0 


68 49-0 137 38-0 (min.) 


84 44 (max.) 


54 


72 26 


65 16 


137 42 


84 36 


55 


76 8 


61 50 


137 58 . 


84 4 


56 


79 44 


58 38 


138 22 


83 16 


57 


83 16 


55 40 


138 56 


82 8 


58 


86 40 


52 48 


139 28 


81 4 


59 


90 2 


50 14 


140 16 79 28 


60 


93 20 


47 46 


141 6 77 48 


70 


124 


28 


152 


56 


80 


152 26 


13 8 


165 34 


28 52 


90 


180 





180 






:1'50 



:1'46 



TOTAL INTERNAL REFLECTION. TABLE II. CONTINUED. 



Col. 1 


Col. 2 


Col. 3 


Col. 4 


Col. 5 


' 


>A and \ calculated only 
to the nearest minute, 
except in the case of 
their equality 


2(*+ x ); being actual 
phasal back-set of 
vibrations in, rela- 
tively to vibrations 
perpendicular to, 
P, produced by one 
reflection 


360'-4(^+ x ); being 
virtual phasal ad- 
vance of vibrations 
in, relatively to vi- 
brations perpen- 
dicular to, P, pro- 
duced by two reflec- 
tions 


H 


2X 


41 28'-3 


0' 


180 0' 


180 0' 


0' 


42 


14 40 


147 16 


161 56 


36 8 


43 


25 6 


126 8 


151 14 


57 32 


44 


32 30 


112 46 


145 16 


69 28 


45 


38 38 


102 46 


141 24 


77 12 


46 


44 


94 42 


138 42 


82 36 


47 


48 54 


87 42 


136 36 


86 48 


48 


53 30 


82 4 


135 34 


88 52 


49 


57 46 


76 58 


134 44 


90 32 


50 


61 50 


72 26 


134 16 


91 28 


51 


65 44 


68 20 


134 4 


91 52 


51 20-41 


67 1-7 


67 1-7 


134 3-4 (min.) 


91 53-2 (max.) 


52 


69 30 


64 36 


134 6 


91 48 


53 


73 6 


61 12 


134 18 


91 24 


54 


76 38 


58 2 


134 40 


90 40 


54 37 








90 


55 


80 6 


55 6 


135 12 


89 36 


56 


83 26 


52 22 


135 48 


88 24 


57 


86 44 


49 48 


136 32 


86 56 


58 


89 58 


47 22 


137 20 


85 20 


59 


93 8 


45 4 


138 12 


83 36 


60 


96 16 


42 54 


139 10 


81 40 


70 


125 40 


25 22 


151 2 


57 56 


80 


154 4 


11 32 


165 36 


28 48 


90 


180 





180 





38 40-9 





180 


180 





39 


10 48 


152 48 


163 36 


32 48 


40 


22 12 


126 40 


148 52 


62 16 


41 


29 36 


111 50 


141 26 


77 8 


42 


35 38 


101 6 


136 44 


86 32 


43 


40 56 


92 36 


133 32 


92 56 


44 


45 42 


85 40 


131 22 


97 16 


45 


50 4 


79 48 


129 52 


100 16 


46 


54 18 


74 38 


128 56 


102 8 


47 


58 14 


70 6 


128 20 


103 20 


48 


62 


66 4 


128 4 


103 52 


48 33-0 


64 0-6 


64 0-6 


128 1-2 (min.) 


103 57-6 (max.) 


49 


65 38 


62 24 


128 2 


103 56 


50 


69 6 


59 6 


128 12 


103 36 


51 


72 34 


56 2 


128 36 


102 48 


52 


75 54 


53 14 


129 8 


101 44 


53 


79 6 


50 38 


129 44 


100 32 


54 


82 18 


48 10 


130 28 


99 4 


55 


85 24 


45 52 


131 16 


97 28 


56 


88 28 


43 42 


132 10 


95 40 


57 


91 30 


41 40 


133 10 


93 40 


58 


94 30 


39 42 


134 12 


91 36 


68 44 








90 


59 


97 26 


37 52 


135 18 


89 24 


60 


100 20 


36 6 


136 26 


87 8 


70 


128 2 


21 34 


149 36 


60 48 


80 


154 18 ! 10 12 


164 30 


31 


90 


180 





180 






FRESNEL'S RHOMB. 



393 



158 rl . That most exquisite invention, Fresnel's rhomb, Mola 
produces a difference of phase of 90 between vibrations in, and 
vibrations perpendicular to, P. By 158"' we see that this can 
only be with glass of refractive index greater than 1'496. Thus, 
the table for refractive index 1*46 shows 84 44' for the maximum 
value in Column 5. The other three tables show, for refractive 
indices To, 1*51, and 1'6 respectively, maximum values in Column 5, 
9028'-8, 9153'-2, and 10357'-6. In each of these cases, there 
are of course two incidences which make the difference of phase 
exactly 90'. Fresnel pointed out* that the larger of the two is to 
be preferred to the smaller; because, with the larger, the differ- 
ences of refractivity, for the different constituents of white light, 
make less error from the desired 90 difference of phase. The 
larger of the two incidences is to be preferred, also, because small 
differences from it make less error on the 90 phasal differences, 
than equal differences from the other. For the same reason, the 
less the refractive index exceeds 1*49G, the better. Fresnel's first 
rhomb was made of glass, of refractive index I'ol, from the 
factory of Saint Gobain : our tables show that 1*5 would reduce 
the phasal errors to about half those given by I'ol, for equal 




Fig. 9'. 

small errors in the direction of the ray. It would thus give an 
appreciably better instrument than Fresnel's own. The accom- 
panying diagram, figure 9', represents a section perpendicular to 

* Collected Papers, Vol. i. p. 792; original date 1823. 



394 LECTURE XVIII. 

.raloM the traversed faces and the reflecting surfaces of the rhomb, for 
refractive index 1'51, as cut by Fresnel; .with acute angles of 
54 37', so that the rays for its proper use may enter and leave 
perpendicularly to the traversed faces. If p denote the perpen- 
dicular distance between the two long sides of the parallelogram, 
the length of the long side is 2ptant (or p . 2'816 with i = 54 37'): 
the length of the short side is p . cosec i (or p . 1 '226). The 
perpendicular distance between the entering and the leaving rays 
is p. 2 sin i (or ^.1-6306). 

158 vii . The length of the long side is chosen such that a 
ray, entering perpendicularly* through the centre of one face, 
passes out perpendicularly through the centre of the parallel face. 
Fresnel's diagram is far from fulfilling this condition; and so are 
many of the diagrams in text-books, and scientific papers on the 
subject. The last diagram I have seen shows only about a quarter 
of normally entering light to suffer two reflections ; and rather 
more than half of it to pass straight through without any reflection 
at all. Airy's and Jamin's diagrams are correct and very clear. 
Fresnel must, in 1823, have given clear instructions to his work- 
man (with or without a diagram); and, to this day, opticians make 
Fresnel's rhombs of right proportions to fulfil the proper condition 
in respect to the entering and leaving rays. But twenty years 
after Fresnel's invention we learn from MacCullaghf that Fresnel 
rhombs were made by Dollond (and probably also by other 
opticians ?), with Fresnel's 54 37' for the acute angle, but with 
refractive indices differing from his 1'51. I am assured that 
some opticians of the present day make the acute angle correct 
according to the refractive index of the glass, to give exactly 90 
phasal difference of the components of the normal ray. I do not 
know if they realize the importance of having glass of refractive 
index as little above 1-496 as possible. 

158 viii . MacCullagh appreciated the beauty and value of 
Fresnel's rhomb, and as early as 1837 had begun using it for 
research. But he was at first much perplexed by unexpectedly 
large errors, until he found means of taking them into account, 

* This, for brevity, I call a " normal ray." 

t " On the Laws of Metallic Reflection and the mode of making experiments upon 
Elliptic Polarization," Proc. Royal Irish Academy, May 8, 1843. MacCullagh's 
Collected Papers, p. 240. 



ERRORS FOUND BY MACCULLAGH. 



395 



";uul of making the rhomb itself serve to measure and to eliminate Molar, 
"them." He good-naturedly adds; "The value of the rhomb as 
"an instrument of research is much increased by the circumstance 
"that it can thus determine its own effect, and that it is not at all 
"necessary to adapt its angle exactly to the refractive index of the 
"glass." This proves a very forgiving spirit: perplexity and loss 
of time in his research gratefully accepted in consideration of his 
having been led to an enlarged view of the value of the Fresnel 
rhomb as an accurate measuring instrument ! 

158 U . MacCullagh had two rhombs from the same maker 
each "cut at an angle of 54^ as prescribed by Fresnel." He found 
one of them wrong phasally by 3, the other by 8 ! He does not 
say whether the errors were of excess or defect ; but we see that 
they must have been excess above 90, because no refractive index 
greater than 1 '5, with t = 54 37', gives as large a defect below 90, 
as 3. This we see by looking at the following Table, calculated 













360 


Excess 


M 




2* 


2 X 


2(* + x) 




above 90 


1-5 


54 37' 


78 4' 


57 28' 


135 32' 


88 56' 


-1 4' 


1*1 




78 48 


56 12 


135 


90 





1-6 




84 14 


46 44 


IHil K 


98 4 


+ 8 4 


1-7 




98 :;-> 


39 4 


127 36 104 48 


14 48 


1-8 





91 44 


33 20 


125 4 


109 52 


19 52 



according to 158 T with t = 54 37', for four refractive indices, 
other than Fresnel's I'ol. We also see that the refractive indices 
of the rhombs, supplied by Dollond to MacCullagh, must have 
been between I'ol and 1*6. 

158*. Notwithstanding MacCullagh's good-natured remark, 
it is important that the acute angle of a Fresnel rhomb should be 
made, as accurately as possible, such that the phasal difference 
shall be exactly 90, for light of definitely specified period (sodium 
light for example), when the direction of the ray is exactly per- 
pendicular to the entering and leaving faces. But however 
trustworthy may be the instrument-maker's work, MacCullagh's 
principle of determining the error in the practical use of the 
instrument, and eliminating it if it is perceptible, is highly im- 
portant and interesting. It may be carried out, either, as he did 



396 LECTURE XVIII. 

Molar, it himself, by means of observations on metallic reflection : or, 
as he suggested, by observations on total reflection at a sepa- 
rating surface of glass and air; a much simpler subject than 
metallic reflection. Or, as simplest method when two Fresnel's 
rhombs are available, we may place them with leaving face of 
one, and entering face of the other, parallel and close together, 
and mount the two rhombs so as to turn independently round 
a common axis perpendicular to these surfaces through their 
centres. The arrangement is completed by mounting two nicols 
so as to turn independently, one of them round the central 
ray entering the first rhomb, and the other round the central ray 
leaving the second rhomb : and providing three graduated circles, 
by which differences of angles turned through between the first 
nicol and first fresnel; between the two fresnels; and between the 
second fresnel and the second nicol ; may be measured. The 
mechanism to do this is of the simplest and easiest. The experi- 
ment consists in letting light ; entering through the first nicol 
and traversing the two fresnels and the second nicol ; be viewed 
by an eye seeing through the second nico). The best approach to 
extinction that can be had, is to be produced by varying the 
three measured angles. For simplicity we may suppose that the 
three zeros of the three pointers, on the three circles, are set so 
that, using for brevity facial intersections to denote intersections 
of the traversed faces and reflecting surfaces of a fresnel, we have 
as follows : 

(1) When the first index is at zero on the first circle, the 
vibrational lines of the light emerging from the first nicol are 
perpendicular to the facial intersections of the first fresnel. 

(2) When the second pointer is at zero on its circle, the facial 
intersections of the first fresnel are parallel to those of the second. 

(3) When the third index is at the zero of the third circle, 
the facial intersections of the second fresnel are perpendicular to 
the vibrational lines of light entering the second nicol. 

Thus if ?i,, a, n 2 denote the readings on the three circles; n 1 is 
the inclination of the vibrational lines of the first nicol to the 
reflectional plane* of the first fresnel; a is the inclination of the 
facial intersections of the first fresnel to those of the second ; and 

* For brevity I call "reflectional plane" of a fresnel, the plane of reflection of a 
normal ray. It is perpendicular to the facial intersections. 



ERRORS OF TWO FRESNELS DETERMINED. 



397 



a is the inclination of the reflectional plane of the second fresnel to Molar, 
the vibrational lines of the light emerging from it. The diagram, 
figure 9", shows these angles all on one circle, ideally seen by an eye 
looking through the second uicol. OF,, OF ly OF*, OV. 2 , represent 
planes through the central emergent ray respectively parallel to the 
vibrational lines of the first nicol, the reflectional plane of the 




Fig. 9". 

first fresnel, the reflectional plane of the second fresnel, and the 
vibrational lines of the second nicol set to let pass all the light 
coming from the second fresnel. OK lt OK, represent planes 
through the common axis perpendicular to OF^ and OF* 

158*'. Let now sin t denote the displacement at time t, of a 
point of ether in the light passing from the first nicol to the first 
fresnel. This implies a special unit of time, convenient for the 
occasion, according to which the period of the light is 2-Tr ; and it 
takes as unit of length, the maximum displacement of the ether 
between the first nicol and the first fresnel. Following the light 
through the apparatus; first resolve the displacement into two 
components, cos n, sin t, sin r^ sin t, parallel to OF l and OK^. The 
phase of the former of these is advanced 90 -f e^ relatively to the 
latter by the two total reflections in the first fresnel, if e^ denote 
its error, which ought to be zero. Thus, at any properly chosen 
point in the space traversed by the light between the two fresnels, 
the displacement of the ether at time t is 

cos n, cos (t + e t ) parallel to OF } ; sin 71, sin t parallel to OK l . 



398 LECTURE XVIII. 

Molar. Resolving each of these into two components, we have, for the 
light entering the second fresnel, 

cos a cos ri! cos (t 4- e^) sin a sin ^ sin t parallel to OF 2 . . .(98 Ji ) ; 
sin a cos n^ cos (t + e^) + cos a sin ?ij sin t parallel to OK^. . .(98 xii ). 

The former of these is advanced 90 + e% relatively to the latter, 
by the two reflections in the second fresnel, e. 2 being its error ; and 
thus, at any properly chosen point in the light passing from the 
second fresnel to the second nicol, the displacement of the ether 
at time t is 
cos a cos rij sin (t + e l + e. 2 ) sin a. sin n^ cos (t + e 2 ), 

parallel to 0^ 2 ...(98 xiii ); 
sin a cos Wj cos (t + e^) + cos a sin 71,. sin t, parallel to OK^ . . .(98 xiv ). 

We have now to find the condition that the resultant of these 
shall be rectilineally vibrating light. This simply implies that 
(98 xiii ) and (98 xiv ) are in the same phase ; and, OF 2 being the line 
of the resultant vibration, we have 

-, = ! = - cot 2 ....(98-), 

where A, B and A', B' denote the coefficients found by reducing 
(98 xili ) and (98 liv ) to the forms A cos t + B sin t, and A' cos t + B' sin t. 
Thus, performing the reductions, and putting 

tanrz 1 = /j 1 , tan?i 2 = /* 2 , tana=j (98 Ivi ), 

we find 

1 _ sin (e l + e 2 ) hj cos e. 2 _ cos (e l + e 2 ) + AI j sin e z 
h^ j cos 6j j sin e l + A x 

(98* vil ). 

These are two equations for finding the two unknowns, e l ,e< i \ when 
hi, h<z,j, are all known; one of them being chosen arbitrarily, and 
the other two determined by observation, worked to produce 
extinction of the emergent pencil (compare 95, 95', above). OV 2 
in the diagram (fig. 9") is perpendicular to the position of the 
vibrational line of the second nicol when set for extinction. 

158 xli . The experimenter will be guided by his mathematical 
judgment, or by trial and selection, to get the best conditioned 
values of h ly h^,j, for determining 61, e a - We must not take Aj = 1 
(that is to say, we must not set the entering nicol fixed with its 
vibrational lines at 45 to the principal planes of the first fresnel) 



SIMPLIFICATION WHEN ERRORS ARE SMALL. 399 

because in the case 4 = (the first fresnel true), the light entering Molar. 
the second fresnel would be circularly polarized, and therefore the 
turning of the second fresnel would have no effect on the quality 
of the emergent light; which would be plane polarized if, and only 
if, this fresnel, like the first, is true (compare 98 above). We 
must not fix h+ = 0, because this would render the first fresnel 
nugatory. It occurs therefore to take fy = 22 which makes 

^-'-^ = 2 and h, = -4142 ............ (98" m ). 

Probably this may be found a good selection if, for any reason, it 
is thought advisable to fix h^ and leave j variable. 

If !, , are so small that we may neglect ef, ej, e^, (98 xvl1 ) 
becomes 



"< 98 *'>= 

whence we have two simple equations for d, e 2 



From these we see that, as e^ and e, are small, the observation must 
make A, and h t each very nearly unity unless j is taken either very 
small or very great. It may be convenient to fix/= 1, (a= 45), 
and to find n lt w 2 by experiment. 

1 58 11 ". When both the fresnels are perfectly true (e 1 =0, e a =0), 
formula (98") shows that A, = h l , if j = ; and h 2 = l/h lt if j = oo : 
but if j has any value between and oo , we must have n, = to, = 45, 
while a may have any value. That is to say, the first fresnel 
produces exactly circular orbits and the second rectifies them. 
When w, is any angle between and 90, we have elliptically 
or circularly polarized light, represented by (98 xl ) and (98^') with 
e, = 0. This is light leaving the first and entering the second 
fresnel. In passing through the second fresnel it becomes con- 
verted into rectilineally vibrating light, represented by (98^") and 
(98 xiT ) with ^ = and e 2 = 0. Thus we see that 158 X 158 xU , 
with e l = and e t = passim, expresses simply arid fully the theory 
of the conversion of rectilineally vibrating light into elliptically 
or circularly polarized light, and vice versd ; by one true Fresnel's 
rhomb. A diagram with rules as to directions in the use of 
Fresnel's rhomb is given below in 158 XV ". 9098 above 



400 LECTURE XVIII. 

Molar, contain the application of the theory to ordinary reflection at 
the surfaces of transparent solids, or liquids, or metals : the first 
fresnel of 158* being done away with, and the reflection substi- 
tuted for it. The corresponding application to total internal 
reflection is much simpler, because the intensities of incident 
and reflected components are equal. It allows the difference of 
phase, produced by the reflection, to be measured with great 
accuracy ; and a careful experimental research, thus carried out, 
would no doubt prove the difference of phase, produced between 
the two components by the reflection, to agree very accurately 
with the obtuse angles calculated for different incidences according 
to (98 vi ) and 158'". 

158 xiv . About a year ago, in making some preliminary 
experiments by aid of a Fresnel's rhomb, to illustrate 90 100, 
152, 153, I interpreted the phasal difference of the rhomb ac- 
cording to Airy's Tracts ; but found error, or confusion, in respect 
to phasal change by one internal reflection in glass, and by metallic 
reflection. I looked through all the other books of reference 
and scientific papers accessible to me at the time; and I have 
continued the inquiry to the present time, by aid of the libraries 
of the Royal Society of London, and the University of Glasgow ; 
but hitherto without success, in trying to find an explicit state- 
ment as to which of the two components is advanced upon the 
other in the Fresnel's rhomb. I have therefore been obliged to 
work the problem out myself mathematically for the Fresnel's 
rhomb ; and with the knowledge thus obtained, to find by experi- 
ment which of the two components is advanced on the other by 
metallic reflection. Of all the authors I have hitherto had the 
opportunity of studying, only Airy in respect to Fresnel's rhomb, 
and Jamin, and Stokes *, and Basset in respect to metallic re- 
flection, have explicitly stated which component of the light is 
advanced in phase. 

In Airy's Tracts, 2nd edition (1831), page 364, I find a 
thoroughly clear statement, agreeing with Fresnel's own, regard- 
ing the Fresnel's rhomb and total internal reflection : " If the 
"light be twice reflected in the same circumstances and with the 
" same plane of reflection, the phase of vibrations in the plane of 
" incidence is more accelerated than that of the other vibrations 

* Mathematical and Physical Paper*, Vol. n. p. 360. 



EIGHTY YEARS' ERROR REGARDING TOTAL REFLECTION. 401 

" by 90." In truth, the vibrations in the plane of incidence are Molar, 
not advanced 90, but set back 270. They are therefore virtually 
advanced 90, relatively to the vibrations perpendicular to the 
plane of incidence. 

In Basset's Physical Optics (Cambridge 1892), page 339, 1 find, 
stated as a law arrived at by Jamin by experiment, with reference 
to metallic reflection, the following : 

" (1) Tlie wave which is polarized perpendicularly to the plane 
" of incidence, is more retarded than that which is polarized in the 
"plane of incidence" 

By experiment I have verified this, working with a Fresnel's 
rhomb, interpreted according to Airy, and 158 V| above. In reading 
Jamin's experimental paper I had felt some doubt as to his mean- 
ing because his expression " vibrations polarisees dans le plan de 
1'incidence " (I quote from memory) may have signified, not that 
the plane of polarization*, but that the line of vibration, was 
in the plane of incidence. That Basset's interpretation was 
correct is however rendered quite certain by a clear statement in 
Jamin's Cours de Physique, Vol. II. page 690, describing relative 
advance of phase of vibrations perpendicular to the plane of inci- 
dence of light reflected from a polished metal. This phasal relative 
advance he measured by a Babinet's compensator, and found it to 
increase from zero at normal incidence, to 90 at the principal 
incidence, 7; and up to 180 at grazing incidence. He would 
have found not advance, but back-set, if he had used a Fresnel's 
rhomb interpreted according to the mathematical theory given (by 
himself as from Fresnel) in pages 783 to 787 of the same volume, 
with the falsified formulas (98 XTli ). 

158 XV . The origin of the long standing mistake regarding 
the Fresnel's rhomb is to be found in Fresnel's original paper, 
" Mdmoire sur la loi des modifications que la reflexion imprime 
"a la lumiere polariseV': reproduced in Volume I. of Collected 
Papers, Paris 1876, pages 767 to 799. In page 777 we find 

* Considering the inevitable liability to ambiguity of this kind, I have abandoned 
the designation ' plane of polarization " ; and have resolved always to specify or 
describe with reference to vibrational lines. Abundant examples may be found in 
the earlier parts of the present volume illustrating the inconvenience of the desig- 
nation "plane of polarization." In fact "polar "and " polarization " were, as is 
now generally admitted, in the very beginning unhappily chosen words for 
differences of action in different directions around a ray of light. These differences 
are essentially not according to what we now understand by polar quality. 
T. L. 26 



402 LECTURE XVIII. 

Molar. Fresnel's own celebrated formulas, for reflected vibrational ampli- 
tudes and their ratio, correctly given as follows : 

tan(i-,i) sin (i - ,i} [ cos (t - ,i) mgxxu 

tan (i + ,i) ' sin (i + ,i) ' cos (i + ,i) ' ' 

It is obvious that the first two of these expressions must have 
the same sign, because at very nearly normal incidences the 
tangents are approximately equal to the sines, and at normal 
incidences, the two formulas mean precisely the same thing ; 
there being, at normal incidence, no such thing as a difference 
between vibrations in, and vibrations perpendicular to, a plane of 
incidence. Yet, notwithstanding the manifest absurdity of giving 
different signs to the "tangent formula" and the "sine formula" 
of Fresnel, we find in a footnote on page 789 (by Verdet, one of 
Fresnel's editors), the formulas changed to 

~4. -%= (8"). 



in consequence of certain " considerations " set forth by Fresnel 
on pages 788, 789. I hope sometime to return to these " con- 
side'rations " and to give a diagram showing the displacements of 
ether in a space traversed by co-existent beams of incident and 
reflected light, by which Fresnel's " petite difficulte " of page 787 
is explained, and the erroneous change from his own originally 
correct formulas is obviated. The falsified formulas (98 xxii ) have 
been repeated by some subsequent writers ; avoided by others. 
But, so far as I know, no author has hitherto corrected the conse- 
quential error, which gives an acute angle instead of an obtuse 
angle for phasal difference in one total internal reflection; and gives 
90 phasal difference instead of 270 in the two reflections of the 
Fresnel's rhomb ; and gives 90 back-set, instead of the truth 
which is 90 virtual advance, of vibrations in the plane of incidence, 
relatively to vibrations perpendicular to the plane of incidence, in 
a Fresnel's rhomb. The serious practical error, in respect to which 
of the two components experiences phasal advance in the Fresnel's 
rhomb, does not occur in any published statement which I have 
hitherto found. Airy accidentally corrected it by another error. 
All the other authors limit themselves to saying that there is a 
phasal difference of 90 between the two components, without 
saying which component is in advance of the other. 



ORBITAL DIRECTION IN LIGHT FROM ANY MIRROR. 403 

_|158 xvi . In Fresnel (page 782), and Airy (page 362), where Molar. 
J 1 is first introduced with the factor \j(p?s\tfi 1), the positive 
sign is taken accidentally, without reference to any other con- 
sideration. A reversal of sign, wherever J 1 occurs in the 
subsequent formulas, would have given phasal advance instead 
of phasal back-set, or back-set instead of advance, in each con- 
clusion. If the authors had included the refracted wave (Airy, 
page 358) in the new imaginary investigation so splendidly dis- 
covered by Fresnel, they would have found it necessary, either to 
reverse the sign of V-l throughout, or to reverse the interpreta- 
tion of it in respect to phasal difference, given as purely con- 
jectural by Fresnel, and by Airy (page 363) quoting from him : 
because with this interpretation, and with the signs as they stand, 
they would have found for the " refracted wave," a displacement 
of the ether increasing exponentially, instead of diminishing 
exponentially, with distance from the interface (see footnote on 
158" above). It is exceedingly interesting now to find that an 
accidentally wrong choice of signs, in connection with V 1, 
served to correct in the result of two reflections, the practical 
error of acute instead of obtuse for the phasal difference due 
to one reflection, which is entailed by the deliberate choice of 
a false sign in the real formulas of (98 xxii ) : and that, thus led, 
Airy gave correctly the only statement hitherto published, so far 
as I know, as to which of the two components experiences phasal 
advance in Fresnel's rhomb. 

158 xvil . Note on circular polarization in metallic or vitreous 
or adamantine reflection. Referring to Basset's Physical Optics, 
page 329, edition 1892, I find Principal Azimuth defined as the 
angle between the plane of polarization of the reflected light, 
and the plane of the reflection, when the incident light is circularly 
polarized light incident at the angle of principal incidence. This 
really agrees w^th the, at first sight, seemingly different definition 
of Principal Azimuth, given in 97" above: because 'it is easily 
proved that when rectilineally vibrating light, is converted into 
circularly polarized light by metallic or other reflection ; the 
azimuth of the vibrational plane of the incident light, is equal to 
the azimuth of the plane of polarization of the reflected light 
when circularly polarized light is converted into rectilineally 
vibrating light by reflection on the same mirror. The fact that k 

262 



404 LECTURE XVIII. 

Molar, is positive for every case of principal incidence ( 97') is included 
and interpreted in the following statement : 

Consider the case of plane polarized light incident on a 
polished metallic, or adamantine, or vitreous, reflector. For sim- 
plicity of expressions let the plane of the reflector be horizontal. 
To transmit the incident light let a nicol, mounted with its axis 
inclined to the vertical at any angle i, carry a pointer to indicate 
the direction of the vibrational lines of the light emerging from it, 
and incident on the mirror. First place the pointer upwards 
in the vertical plane through the axis ; which is the plane of 
incidence of the light on the mirror. The reflected light is of 
rectilineal vibrations in the same plane. Now turn the pointer 
anti-clockwise through any angle less than 90. The reflected 
light consists of elliptic, or circular, anti-clockwise orbital motions. 

If i = I, the principal incidence ; the two axes of each elliptic 
orbit are, one of them horizontal and perpendicular to the plane 
of incidence and reflection; the other in this plane. 

To avoid any ambiguity in respect to " clockwise " and " anti- 
clockwise," the observer looks at the nicol, and at the circle in 
which its pointer turns, from the side towards which the light 
emerges after passing through it : and he looks ideally at the 
orbital motion of the reflected light from the side towards which 
the reflected light travels to his eye. See 98 above. 

In external reflection of rectilineal ly vibrating light by all 
ordinary transparent reflectors, including diamond (but not realgar), 
the deviation from rectilineality of the reflected light is small, 
except for incidences within a few degrees of the Brewsterian 
tan" 1 /i. See diagram of 178 below (figures 11 and 12), for 
diamond. 

158 xviii . The rules for directions in elliptic and circular 
polarization by a Fresnel's rhomb are represented by the annexed 
diagram, figure 9'". 0' and are the centres of the entering and 
exit faces. OK is a line parallel to the facial intersections. OF 
represents the plane of the reflections in the rhomb, being a plane 
perpendicular to the four optically effective faces. 

Cases ttj, a 2 , a 3 . Plane polarized light enters by 0'. OVi, 
OF 2 , OV 3 , are parallels through to the vibrational lines of the 
entering light ; they are of equal lengths, to represent the displace- 
ments as equal in the three cases. The orbits of the exit light in 



RULES FOR ORBITAL DIRECTION. 



405 



the three cases are represented by (1) ellipse, (2) circle, (3) ellipse; Molar, 
being drawn according to the geometrical construction indicated 
in one quadrant of the diagram. 

Cases 6,, 6 a , b t . The orbits in three cases of equally strong 
circularly and elliptically polarized entering light, having axes 
along OF and OK, shown with their centres transferred to 0, are 
represented by (1) ellipse, (2) circle, (3) ellipse. OF/, OF,', OF,', 
represent the displacements in the exit light, which in each of 
these three cases consists of rectilineal vibrations. 




Fig. 9'". 

The direction of the orbital motions of the exit light in cases a, 
and of the entering light in cases 6, is, in each of the six cases, 
anti-clockwise, as indicated by the arrowheads on the ellipses and 
circle. 

Thus we have the following two rules for directions : 

Rule (a). When the vibrational plane of entering light must 



406 LECTURE XVIII. 

Molar, be turned anti-clockwise to bring it into coincidence with OK, 
the orbital motions in the exit light are anti-clockwise. 

Rule (6). When the orbital motion of circularly or elliptically 
polarized entering light, having axes parallel to OK and OF, is 
anti-clockwise, the vibrational plane of the exit light is turned 
anti-clockwise from OK. 

These rules, and the diagram ; hold with a "quarter-wave 
plate" substituted for the fresnel, provided homogeneous light of 
the corresponding wave-length is used. The principal plane of 
the quarter-wave plate in which are the vibrations of waves of 
greatest propagational velocity, is the OF of the diagram. 

159. The beautiful discovery made seventy years ago by 
MacCullagh and Cauchy, that metallic reflection is represented 
mathematically by taking an imaginary complex to represent the 
refractive index, p., still wants dynamical explanation. In 1884 
we saw (Lee. XII. pp. 155, 156) that yu, 2 is essentially real and 
positive through a definite range of periods less than any one of 
the fundamental periods, according to the unreal illustrative mech- 
anism of unbroken molecular vibrators constituting the ponderable 
matter; embedded in ether, and acting on it only by resistance 
between ether and the atoms against simultaneous occupation of 
the same space. This gave us a thoroughly dynamical foundation 
for metallic reflection in the ideal case of no loss of light, and for 
the transmission of light through a thin film of the metal with 
velocity (as found by Quincke), exceeding the velocity of light in 
void ether. It however gave us no leading towards a dynamical 
explanation of the manifestly great loss of light suffered in reflection 
at the most perfectly polished surfaces of metals other than silver 
or mercury (see 88 above). But now the new realistic electro- 
ethereal theory set forth in Appendices A and E, and iu 162... 168 
below, while giving for non-conductors of electricity exactly the same 
real values of p?, negative and positive, as we had from the old 
tentative mechanism, seems to lead towards explaining loss of 
luminous energy both in reflection from, and in transmission 
through, a substance which has any electric conductivity, however 
small. In App. E 30, J. J. Thomson's theory of electric conduction 
through gases is explained by the projection of electrions out of 
atoms. If this never took place, the electro-ethereal theory would, 
like our old mechanical vibrator, give loss of energy from trans- 
mitted light only in the exceedingly small proportion due to tho 



PERFECT OPACITY WITH PERFECT REFLECTIVITY. 407 

size of the atoms according to Rayleigh's theory of the blue sky Molecular. 
( 58 above). On the other hand the crowds of loose electrions 
among under-loaded atoms, throughout the volume of a metal, by 
which its high electric conductivity is explained in App. E 30, may 
conceivably give rise to large losses of energy from reflected 
light; losses spent in heating a very thin surface layer of the 
metal by irregular motions of the electrions. This seems to me 
probably the true dynamical explanation of the imaginary term 
in i/ 3 of 158 (98). See also 84 above. 

159'. In every case of practically complete opacity (that is 
to say, no perceptible translucency from the most brilliant light 
falling on a plate of any thickness greater than three or four 
wave-lengths, or about 2 . 10~ 4 cm.), measurement of the principal 
azimuth, which can be performed with great accuracy by aid of 
two nicols and a Fresnel's rhomb ( 97 97" above), gives an 
interesting contribution to knowledge regarding loss of luminous 
energy in the reflection of light. The notation of 94' gives 

S* sin 2 a + (T 2 + E*) cos 1 a ............... (98"') 

as the reflectivity for light .vibrating in azimuth a from the 
plane of incidence ( 88, foot-note); so that in every case of 
practical opacity, 

1 - [& sin 2 a + (T- + E*) cos a] ............ (98" 11 ) 

represents loss of luminous energy by conversion into heat in a 
thin surface stratum of less than 2 . 10~ 4 cm. thickness. 

159". For perfect reflectivity (98 Mii ) must be zero for every 
value of a, and for every incidence. Hence, as S, T, E are inde- 
pendent of a, we have, at every incidence, 



Hence for the incidence making T=Q (that is the principal 
incidence), S' = E* = 1 ; which makes E/S = 1 in every case of 
perfect reflectivity, in total internal reflection for instance. There- 
fore the principal azimuth, being tan" 1 (E/S) for principal inci- 
dence, is 45, if the reflectivity is perfect. (See 157 above.) 

159'". Observation and mathematical theory agree that the 
principal azimuth is positive in every case : for interpretation of 
this see 158 xvli above. They also agree that in every case short 
of perfect reflectivity the principal azimuth is < 45, not > 45. 



LECTURE XIX. 



FRIDAY, October 17, 3.30 P.M., 1884. Written afresh, 1903. 



Reconciliation between Fresnel and Green. 

Molecular. THIS Lecture or " Conference " began with the consideration 
of a very interesting report, presented to us by one of our twenty- 
one coefficients, Prof. E. W. Morley, describing a complete solu- 
tion, worked out by himself, of the dynamical problem of seven 
mutually interacting particles, which I had proposed nine days 
previously (Lecture IX. p. 103 above) as an illustration of the 
molecular theory of dispersion with which we were occupied. 
His results are given in the following Table. 

Solution for Fundamental Periods, Displacement, and Energy 
Ratios of a System of Spring-connected Particles. By EDWARD 
W. MORLEY, Cleveland, Ohio. 

/i=l, 4, 16, 64, 256, 1024, 4096. 
(7=1, 2, 3, 4, 5, 6, 7, 8. 

Fundamental Periods corresponding to outer ends of 
Springs 1 and 8 held fixed. 













I 




T 2 = 


2889 


9952 


3-350 . 


11-362 


39-12 


137-89 


680-2 


1 _ 


3-4618 


1-00483 


0-29849 


0-0880078 


0-0255607 


0-0072564 


0-0014701 


1 _ 


1-860 


1-002 


546 


296 


159 


0851 


0383 



















A PROBLEM OF SEVEN VIBRATIONAL MODES SOLVED. 409 

Displacement Ratios, or values of P\ . Molecular. 



*1 


1 


1 


1 


1 1 


1 


1 


x ~. 


-231 


1-000 


1-351 


1-456 1-487 


1-496 


1-499 


X, 


014 


-341 


1-047 


1-589 1-761 


1-813 


1-829 


X 4 


-ra27 


026 


-431 


1-129 1-787 


1-997 


2066 


* 


v!5 


-in 50 


033 


- -511 1-223 


1-960 


2-216 


* 


- -VUI 26 


v30 


-m68 


040 --581 


1-322 


2-203 


*7 


xi IS 


- -vin 51 


v39 


-in81 -045 


-628 


1-717 



Energy Ratios, or values of *l . 

* 



l*l" 


1 


| 


1 


1 


1 


1 


1 


V,' 


213 3-998 


7-30 


8-48 


8-85 


8-96 


8-99 




ii 33 1-864 


17-54 


40-41 


49-64 


52-58 


53-54 


* 4 ' 


v 47 -039 


11-88 


81-66 


204-35 


255-34 


273-14 


*,' 


n61 -iv 65 


28 


66-73 


382-71 


983-10 


1157-52 


,*. 


xiv 7 -TIII 9 


HI 47 


1-62 


345-60 1788-13 4968-41 


%v 


xx 7 


xnl 


vn63 


002 


8-42 


1616-99 


12080-04 


Sam 


1-216 


6-902 


38-0004 


199-90 


1000-57 


4706-10 


18542-64 



At present our subject is the dynamical reconciliation between Molar. 
Fresnel and Green, not only in respect to reflection and refrac- 
tion at an interface between two isotropic transparent bodies, 
as promised in Lecture XVIII. 133, 139, but also in respect to 
the propagation of light through a transparent crystal (double 
refraction) as promised in Lecture XV. 45. 

160. In my paper on the reflection and refraction of light 
(Phil. Mag. 1888, 2nd half-year), an extract from which is quoted 
in 107111 of Lecture XVIII., I have shown ( 109111) 
that a homogeneous portion of an elastic solid with its boundary 



410 LECTURE XIX. 

Molar, h Id fixed is stable if its rigidity (??,) is positive ; even though its 
bulk-modulus (k) is negative if not less than ^n. And in 115 
it is shown that the propagational velocity (v) of condensational- 
rare factional waves in any homogeneous elastic solid (or fluid if 
n = 0) is given by the equation 



In virtue of this equation it had always been believed that the 
propagational velocity of condensational-rarefactional waves in an 
elastic solid was essentially greater than that of equivoluminal 

waves, which is . /-. The Navier-Poisson doctrine, upheld by 

many writers long after Stokes showed it to be wrong (see 
Lecture XI. pp. 123, 124 above, and Appendix I below), made 
lc = |n (see Lecture VI. p. 61 above), and therefore the velocity of 
condensational-rarefactional waves = \/3 times that of the equi- 
voluminal wave. The deviations of real substances, such as metals, 
glasses, india-rubber, jelly, to which Stokes called attention, were 
all in the direction of making the resistance to compression greater 
than according to the Navier-Poisson doctrine ; but it was pointed 
out in Thomson and Tait ( 685) that cork deviates in the opposite 
direction and is, in proportion to its rigidity, much less resistant 
to compression than according to that doctrine. In truth, without 
violating any correct molecular theory, we may make the bulk- 
modulus of an elastic solid as small as we please in proportion to 
the rigidity provided only, for the sake of stability, we keep it 
positive. A zero bulk-modulus makes the velocity of condensational - 
rarefactioual waves equal to \/$ times that of equivoluminal waves. 
But now, to make peace between Fresnel and Green, we want for 
ether; if not all ether, at all events ether in the space occupied by 
ponderable matter, a negative bulk-modulus just a little short of 

f n, to make the velocity (v) very small in comparison to . /-. 

And now, happily ( 167 below), a theory of atoms and electrions 
in ether, to which I was led by other considerations, gives us 
a perfectly clear and natural explanation of ether through void 
space practically incompressible, as Green supposed it to be; while 
in the interior of any ordinary solid or liquid it has a large enough 
negative bulk -modulus to render the propagational velocity of 



RETURN TO HYPOTHESIS OF 1887 WITH MODIFICATION. 411 

roixlensational-rarefactional waves exceedingly small in comparison Molar, 
with that of equivoluminal wavea 

161. In my 1888 Article I showed that if v is very small, 
or infinitely small, in proportion to the propagational velocity of 
equivoluminal waves in the two mediums on the two sides of a 
reflecting interface, or quite zero, Fresnel's laws of reflection and 
refraction are very approximately, or quite exactly, fulfilled. About 
fourteen years later I found that, as said in 46 of Lecture XV., 
it is enough for the fulfilment of Fresnel's laws that the velocity 
of the condensational -rare factional waves in one of the two mediums 
be exceedingly small. During those fourteen years, I had been 
feeling more and more the great difficulty of believing that the 
compressibility-modulus of ether through all space could be nega- 
tive, and so much negative as to make the propagational velocity 
of condensational-rarefactional waves exceedingly small, or zero. 
One chief object of the long mathematical investigation regarding 
spherical waves in an elastic solid, added to Lecture XIV. (pp. 191 
219 above), was to find whether or not smallness of propaga- 
tional velocity of condensational-rarefactional waves through ether 
void of ponderable matter could give practical annulment of energy 
carried away by this class of waves from a vibrator constituting a 
source of light. I found absolute proof that the required practical 
annulment was not possible; and I therefore felt forced to the 
conclusion stated in 32, p. 214: "This, to my mind, utterly 
" disproves my old hypothesis of a very small velocity for irro- 
" tational wave-motion in the undulatory theory of light." Now, 
most happily, seeing that it is enough for the dynamical verification 
of Fresnel's lawo that the velocity of the condensational-rare- 
factional wave be exceedingly small for either one or other of the 
mediums on the two sides of the interface, I can return to my old 
hypothesis with a confidence I never before felt in contemplating 
it. It is, I feel, now made acceptable by assuming with Green 
that ether in space void of ponderable matter is practically incom- 
pressible by the forces concerned in waves proceeding from a source 
of light of any kind, including radiant heat and electro-magnetic 
waves; while, in the space occupied by liquids and solids, it has a 
bulk-modulus largely enough negative to render the propagational 
velocity of condensational-rarefactional waves exceedingly small in 
comparison with that of equivoluminal waves in pure ether. We 



412 LECTURE XIX. 

Molar, have now no difficulty with respect to getting rid of the condensa- 
tional -rare factional waves generated by the incidence of light on a 
transparent liquid or solid. They may, with very feeble activity, 
travel about through solids or liquids, experiencing internal re- 
flections, almost total, at interfaces between solid or liquid and air 
or vacuous ether; but more probably ( 172 below) they will be 
absorbed, that is, converted into non-undulatory thermal motion, 
among the ponderable molecules, without ever travelling as waves 
through more than an exceedingly small space containing ponder- 
able matter (solid, or liquid, or gas). Certain it is that neither 
ethereal waves, nor any kind of dynamical action, within the body, 
can give rise to condensational-rarefactional waves through void 
ether if we frankly assume void ether to be incompressible. 

Dlecular. 162. So far, we have not been supported in our faith by 
any physical idea as to how ether could be practically incom- 
pressible when undisturbed by ponderable matter ; and yet may be 
very easily compressible, or may even have negative bulk-modulus, 
in the interior of a transparent ponderable body. 

Do moving atoms of ponderable matter displace ether, or do 
they move through space occupied by ether without displacing it? 
This is a question which cannot be evaded: when we are concerned 
with definite physical speculations as to the kind of interaction 
which takes place between atoms and ether; and when we 
seriously endeavour to understand how it is that a transparent 
body takes wave-motion just as if it were denser than the 
surrounding ether outside, and were otherwise undisturbed by 
the presence of the ponderable matter. It is carefully considered 
in Appendix A, and in Appendix B under the heading " Cloud I." 
My answer is indicated in the long title of Appendix A, " On the 
"Motion produced in an Infinite Elastic Solid by the motion 
" through the space occupied by it, of a body acting on it only by 
" Attraction or Repulsion." This title contradicts the old scholastic 
axiom, Two different portions of matter cannot simultaneously 
occupy the same space. I feel it is impossible to reasonably gain- 
say the contradiction. 

163. Atoms move through space occupied by ether. They 
must act upon it in some way in order that motions of ponderable 
matter may produce waves of light, and in order that the vibratory 



REAL MOLECULAR EXPLANATION. 413 

motion of the waves may act with force on ponderable matter. Molecula 
We know that ether does exert force on ponderable matter in 
producing our visual perception of light ; and in photographic 
action ; and in forcibly decomposing carbonic acid and water by 
sunlight in the growth of plants; and in causing expansion of 
bodies warmed by light or radiant heat. The title of Appendix A 
contains my answer to the question What is the character of the 
action of atoms of matter on ether? It is nothing else than 
attraction or repulsion. 

164. But if ether were absolutely incompressible and in- 
extensible, an atom attracting it or repelling it would be utterly 
ineffectual. To render it effectual I assume that ether is capable 
of change of bulk ; and is largely condensed and rarefied by large 
positive and negative pressures, due to repulsion and attraction 
exerted on it by an atom and its neutralising quantum of electrions. 
As in Appendix A, 4, 5, I now for simplicity assume that an 
atom void of electrions repels the ether in it and around it with 
force varying directly as the distance from the centre for ether 
within the atom, and varying inversely as the square of the 
distance for ether outside. I assume that a single separate 
electrion attracts ether according to the same laws; the radius 
of the electrion being very small compared with the radius of any 
atom. I assume that all electrions are equal and similar, and 
exert equal forces on ether. 

165. While keeping in view the possibility referred to in 
6 of Appendix E, I for the present assume that the repulsion of 
a void atom* on ether outside it is equal to an integral number of 
times the attraction of one electrion on ether ; same distances 
understood. An atom violating this equation cannot be unelec- 
trified. I continue to make the same assumptions as in Appendix D 
in respect to mutual electric repulsion between void atoms and 
void atoms ; attraction between void atoms and electrions ; and 
repulsion between electrions and electrions. And as in Appendix A 
an " unelectrified atom " is an atom having its saturating quantum 
of electrions within it. 



* For brevity I use the expression " void atom" to signify an atom having no 
electrion within it, 



414 LECTURE XIX. 

Molar. 166. I assume that the law of compressibility of ether is 
such as to make the mean density of ether within any space which 
contains a large number of unelectrified atoms, exactly equal to 
the natural density of ether undisturbed by ponderable matter. 
If there were just one electrioii within each atom this assumption 
would exactly annul displacement of the ether outside an atom by 
the repulsion of the atom and the attraction" of the electrion ; and 
would very nearly annul it when there are two or more electrions 
inside. Thus the ether within each atom is somewhat rarefied 
from the surface inwards: and farther in, it is condensed round each 
electrion. In a mono-electrionic atom, the spherical surface of 
normal density between the outer region of rarefaction and the 
central region of condensation, I call for brevity, the sphere of con- 
densation. In a poly-electrionic atom the density of the ether 
decreases from enormous condensation around the centre of each 
electrion to exactly normal value at an enclosing surface, the space 
within which I shall call the electrion's sphere of condensation. 
It is very approximately spherical except when, in the course of 
some violent motion, two electrions come very nearly together. 

167. I assume that the law of compressibility of ether is 
such as to make the equilibrium described in 166 stable; but so 
nearly unstable that the propagational velocity of coridensational- 
rarefactional waves travelling through ether in space occupied by 
ponderable matter, is very small in comparison with the propa- 
gational velocity of equivoluminal waves through ether undisturbed 
by ponderable matter. 

168. Lastly I assume that the effective rigidity of ether in 
space occupied by ponderable matter is equal to that of pure ether 
undisturbed by ponderable matter. This is not an arbitrary 
assumption : we may regard it rather as a proposition proved 
by experiment, as explained in Lecture I. pp. 15, 16 and 81', 136 
of Lectures XVII. and XVIIT. But, as will be shown in Lee. XX., 
237 below, it is somewhat satisfactory to know that it follows 
directly from a natural working out of the set of assumptions of 
163167. 

169. Return now to 132. This is merely Green's wave- 
theory extended to include not only the equivoluminal waves but 



DYNAMICAL THEORY OF ADAMANTINISM. 415 

also the condensational-rarefactional waves resulting from the Molar, 
incidence of plane equivoluminal waves on a plane interface 
between two sliplessly connected elastic solids of equal rigidities, 
each capable of condensation and rarefaction. Let us suppose 
that in either one or other of the two mediums the propagational 
velocity of the condensational-rarefactional wave is very small. 
This makes L very small as we see by using (48) and (50) to 
eliminate a, c, and ,c from (Gl) which gives 

L= ( *.- 

,p \/(u~ J - sin 1 i 

170. If either the upper or the lower medium be what we 
commonly call vacuum, (in reality ether void of ponderable matter), 
the upper for example, we have v = x , and (100) becomes 

~~ ,pt sin iu~ l + p V(,v~* - sin it*- 1 )' 
whence when ,v is very small 

L = ^ - sin"tM~,v (!02), 

where = denotes approximate equality. If v and ,v are each very 
small (100) becomes 



(103), 



,o,v + pv 

thus by (100), (102) and (103), we see not only, as said above in 
133, that Fresnel's Laws are exactly fulfilled if either v or ,v is 
zero. We see farther how nearly, to a first approximation, they are 
fulfilled if v = oo , and ,v is very small ; also if v and ,v are both 
very small. 

Probably values of ,v, or of v and ,v, as small as w/500 or 1//1000 
may be found small enough, but w/100 not small enough, to give 
as close a fulfilment of Fresnel's Laws as is proved by observation. 
See 182 below. 

171. Thus so far as the reflection and refraction of light are 
concerned, the reconciliation between Fresnel and Green is com- 
plete : that is to say we have now a thoroughly realistic dynamical 
foundation for those admirable laws which Fresnel's penetrating 
genius prophesied eighty years ago from notoriously imperfect 
dynamical leadings. (See 106, 107 above.) 



416 LECTURE XIX. 

Molar. 172. The only hitherto known deviation from absolute 
rigour in Fresnel's Laws for the reflection and refraction of light 
at the surface of a transparent body in air or void ether, or at the 
interface between two transparent bodies liquid or solid, is that 
which was discovered by Sir George Airy for diamond in air more 
than eighty years ago, and many years afterwards was called by 
Sir George Stokes the adamantine property. (See 158 above.) 
It is shown in the Table of 105 above along with corresponding 
deviations discovered subsequently by Jamin in other transparent 
bodies solid and liquid. It now appears by (102) and (103) that 
the explanation of these deviations from Fresnel, which even in 
diamond are exceedingly small, is to be found by giving very small 
imaginary values to t v, or to v and ,v. To suit the case of light 
travelling through vacuum or through air and incident on a trans- 
parent solid or liquid, we should take v = oo if the incident light 
travels through vacuum ; or if through air, v perhaps = oo , but 
certainly very great in comparison with ,v. Hence in either case 
the proper approximate value of L is given by (102). 

173. Looking to (59) and (67) of 132 we see that if ,v have 
a small real value (positive of course) the reflected ray, of vibrations 
in the plane of incidence, will vanish for an angle of incidence 
slightly less than the Brewsterian angle tan" 1 //,. If we give to ,v 
a complex value ,p + i,q with ,p positive, we have the adamantine 
property, and principal incidence slightly less than with ,p = 0. 
Hitherto we have no very searching observations as to the perfect 
exactness of tan" 1 /* whether for the polarizing angle when the 
extinction is seemingly perfect or for the principal incidence in 
cases of perceptible adamantine property. For the present there- 
fore we have no reason to attribute any real part to ,?;; and we 
may take 

y= - ? 2 ; ,v = tq ................... (104), 

where q denotes a real velocity. Using this in the last formula of 
128 and eliminating sin ,j/,v by (48) and cos f j/,v and ,c by (49) 
and (50) we find for the displacement in the refracted couden- 
sational-rarefactional wave 



(105). 



Looking now to (63) and (102) we see that in t H/G there is an 
imaginary part which is exceedingly small in comparison with the 



IMAGINARY VELOCITY OF CONDENSATIONAL WAVES. 417 

real part, because, by (102), L/,v is real and not small, while L is Molar, 
imaginary and exceedingly small. For our present approximate 
estimates we suppose G to be real and therefore ,H to be real. 
Taking now f(t) = e** in (105) and taking the half-sum of the 
two imaginary expressions as given with + 1, we find for the real 
refracted condensational-rarefactional displacement the following 
expression ; with approximate modification due to qa being a very 
small fraction; 

D = ,H cos a (t - ax) e-* l+a VW == ft cos a> (t - ax) e"*"?. . .(106). 
Hence q is positive. (See footnote on 158".) The direction of 
this real displacement is exceedingly nearly OF' of 117, that is 
to say the direction of Y negative, because the imaginary angle ,j 
differs by an exceedingly small imaginary quantity from 90. It 
would be exactly 90 if ,v were exactly zero ; and it would be 
less than 90 by an exceedingly small real quantity if ,v were an 
exceedingly small real velocity. 

174. The motion represented by (106) is not a wave 
travelling into the denser medium : it is a clinging wave travelling 
along the interface with velocity a" 1 . The direction of the displace- 
ment is approximately perpendicular to the interface. Its magni- 
tude decreases inwards from the interface according to the law of 
proportion represented by the real exponential factor; distance 
inwards from the interface being y. The period of the wave is 
2-7T/&) ; and the space travelled in this time with velocity q is Zirq/at. 
Hence the displacement at a distance from the interface equal to 
this space, is 1/e 2 " or 1/535 of the displacement at the interface. 

175. Consideration of the largeness of any such distance as 
2 . 10~ 8 cm. ( 80 above) between centres of neighbouring atoms, 
and of the smallness of t v if real, will probably give good dynamical 
reason for the assumption of ,v a pure imaginary. It is a very 
important assumption, inasmuch as it implies that there is no 
inward travelling condensational-rarefuctional wave carrying away 
energy from the equivoluminal reflected and refracted rays. 

176. Let us now find the value of qfu, the adamantinism 
( 158); to give any observed amount of the adamantine property, 
as represented by the tangent of the Principal Azimuth, which is 
Jamin's k. In (59) of 132 take for L the value given by (102), 

and put ( ' p ~ py * = N; ,v=tq- ^ = a (107), 

T. i, 27 



418 LECTURE XIX. 

Molar, thus we find 

-G' _ u(,pb-p,b)-iN<r8m*i P - 



_ = 

G ~u(,pb + p,b) + iNa sin 2 i ~ 2 ( ,pb + p ,6) 2 + (No- sin 2 i) 3 

............ (108)*, 

where P = u* [( ,pb)* - ( P ,bY] - (No- sin 2 iY\ 

Q=2u,pbNa-sm-i 

Hence taking in 128, f(t) = e" '; and realising by taking half-sum 
of solutions for i, we find displacement 



(s\ 
t j 

in reflected wave = - G J(T- + E-) cos \a> (t - ~\ - < 1 . . .(Ill), 



where < = tan" 1 p = tan" 1 -^ 

....(112). 

_ _j 2/i 2 (/u 2 1) o- cos i sin 2 r 

/i 2 cos* i sin 2 i (/i a 1 ) 3 a- 2 sin 4 i , 

The developed form of Q/P here given is found by putting 

,p/p = fj?; ub = cosi', u l b = ficoa f i' i cos 2 ,i = l ...(113), 

and by dividing numerator and denominator by (/x 2 l)/f*?. 



* According to the notation of 94' we have QjP = EIT, and 

p Q 

T= u* (,pb + p,b)* + (N sin 2 i g/u) and E = u (,pb + p,6) 2 + (N sin a i g/u)* ' 

t This alternative form (111') comes from (111) by resolving P'-' + Q 2 into two 
factors according to the algebraic identity 

(a 2 - fc 2 - c 2 ) 2 + 4a 2 c 2 = [(a + fc) 2 + c 2 ] [(a - fc) 2 + c 2 ]. 

But it is found directly by treating (108) as we treated a similar formula in (75), 
following Green, on a plan which is simpler in respect to the resultant magnitude, 
but less simple in respect to the phase, than the plan of (111). 



ADAMANTINE REFLECTION. 419 

177. By (111') and the notation of 94', we have Molar. 



|_ 
_ ,,(JV 

(, P b + P ,by [ (,pb+ P ,br + (AVsin'tTj ' 

or approximately 

rtan(t~,ty|'_ 4p,p6,6Cy<rsm't)' 
- - - < 



In (114) and (115) T'+ E* denotes the whole intensity of the 
reflected light, due to incident light of unit intensity vibrating 
in the plane of incidence, and the smallness of the right-hand 
members of these equations shows how little this exceeds that 
calculated according to Fresnel's formula 

rtan(t-,tT| 
[_tan ('-- ,t)J 

Compare 103105 above. 




Fig. 10. 

178. In (110) and (111), * denotes a length AEF (fig. 10) 
in the line of the incident ray produced through E the point of 
incidence, or AEF' an equal length in the path of the incident 
and the reflected ray. Thus we see that the reflection causes 
a phasal set-back </> which increases from through 90 to 180 
when the angle of incidence is increased from to 90, as shown by 
(112); because (112) shows that tan <f> increases through positive 
values from to + oc when i increases from to a value slightly 
less than tan" 1 fi : and, when i increases farther up to 90, tan <p 
increases through negative values from - oo to 0. These variations 
are illustrated by the accompanying diagrams (figs. 11, 12) drawn 
according to the following Table of values of tan <f> and <p calculated 

27-2 



420 



LECTURE XIX. 




DIAGRAM OF PHASAL LAG IN ADAMANTINE REFLECTION. 421 

from (112) with a = '00293 as for diamond according to 181 Molar. 
below. Note how very slowly tan <j> and < increase for increasing 



Diamond 



<r= -00293 



30 40 SO 

/ n.e i de n ec* 



Fig. 12. 



1 


tan* 





t 


tan* 














I 


00 


90 


1 


000008798 


0-03' 


67-7 


- 14-87770 


93 51' 


10 


0008890 


03' 


67-75 


- 6'95857 


98 11' 


20 


003678 13' 


67-8 


-4-53967 


102 25' 


30 


008835 30' 


67-9 


-2-87607 


110 29' 


40 


01772 


1!' 


68-0 


- 1-89599 


117 49' 


50 


03469 


159" 


68-l 


-1-46724 


124 17' 


60 


08784 


51" 


68-2 


-1-19615 


129 54' 


65 


25164 


14 7' 


68-3 


-1-00921 


134 44' 


67 


1-00386 


45 7' 


68-4 


- -872514 


138 54' 


67'l 


1-18360 


49 48' 


70 


- -27083 


164 51' 


67 -2 


1 '44216 55 16' 


75 


- -07694 


175 36' 


67'3 


1-85085 61 37' 


80 


- -01640 


179 4' 


67 '4 


2-56306 68 41' 


89 


- -00299 


179 50' 


67'5 


4-20197 76 37' 


90 


00000 


180 O 7 


67 "55 


6-18014 80 49' 








67 '6 11-70150 85 7' 








tan-V-0-00883 = / 90 O 7 









422 LKCTUllE XIX. 

Molar, incidences, except those very slightly less than, and very slightly 
greater than, the Principal Incidence, 67 0> G5. Note also how very 
suddenly tan $ rises from a small positive value to + oo and from 
oo to a very small negative value when the incidence increases 
through the Principal Incidence. Note also how suddenly there- 
fore </>, the phasal retardation, increases from a small positive 
quantity up to 90 when the incidence reaches 67'65 ; and, when 
the incidence increases to slightly above this value, how suddenly 
the phasal retardation rises to very nearly 180. But carefully 
bear this in mind that tan <f> is essentially zero for i = 0, and for 
i = 90, as shown by (112). 

179. The sign minus before G in (111) signifies a phasal 
advance or back-set of 180. For incidence and reflection of rays 
having vibrations perpendicular to the plane of incidence, we have 
also essentially the sign minus before C in (116) below, taken 
from (20) and (23) of 117 and (39) of 123 as corresponding 
to (110) and (111). Thus for amplitudes we have 

(p ' 
t 
u 

b _ b r 

reflected wave C '-, 1 cos 

,0+0 

Here there is neither retardation nor acceleration corresponding to 
the <f> of (111). We infer that if plane polarized light falls on the 
reflecting surface with its vibrational plane inclined at any angle a 
to the plane of incidence, and if we resolve it ideally into two 
components having their vibrational planes in and perpendicular 
to that plane, the component of the reflected light due to the 
former will be phasally behind that due to the latter by the 
retardation <f> given by (112). Hence for every incidence the 
reflected light will be elliptically polarized ; or circularly in case of 
phasal difference 90, and equal amplitudes, of the two reflected 
components. The values of i and of a which make the reflected 
light circularly polarized, are called the "Principal Incidence" 
and " Principal Azimuth " (see 97 above). 

180. The Principal Incidence, /, is the value of i which 
annuls the denominator of E\T as seen in (112). It is therefore 
given by the equation 

tan- /= /*- -O* 2 -!) 3 ^ tan 2 /sin 2 / (117). 




PRINCIPAL INCIDENCE IN ADAMANTINE REFLECTION. 423 

Notwithstanding the greatness of (p? I) 3 for substance of high Molur. 
refractivity (Example 1267 for Sulphuret of Arsenic) the second 
term of the second member of (117) is very small relatively to the 
first because of the smallness of (q/uf for all transparent substances 
for which we have the observational data from Jamin. (See 182 
below.) Hence in that second term we may take /* s for tan 2 / 
and /*'/(/"* + 1) for sin 2 /. Thus instead of (117) to determine the 
Principal Incidence / we have 



181. To find the Principal Azimuth a; let D be the vibra- 
tional amplitude of plane polarized incident light ; and D cos a, 
D sin a the vibrational amplitudes of the components in and 
perpendicular to the plane of incidence. Thus with the notation 
of 176, 179 we may take 

G = Dcosa, 0=Z)sina ............... (119). 

These must be such as to make the two components of the reflected 
light equal when the angle of incidence is that given by (117) 
(the Principal Incidence); that is to say, the value of i which 
makes P ; or approximately according to (109), ,pb = p,b. This 
makes the amplitude of (111) approximately equal to - GE, or 



Equating this to the second line of (116) and using (119), we see 
that the condition for circular polarization is 



Now the angle of incidence is approximately tan" 1 /A; for which 
we have 

k-ji (122), 

and therefore for a, the principal azimuth, we have 

" (123). 



424 



LECTURE XIX. 



Molar. | 182. The Table of 105 above gives Jamin's values of tan a 
(his "k") for eight substances, and the result of a very rigorous 
examination by Rayleigh for water with specially purified surface 
to be substituted for Jamin's which was probably vitiated by 
natural impurity on the water surface. Omitting fluorine because 
of its exceptional negative value for tan a, which if genuine must 
be explained (see 158 above) otherwise than by my assumption 
of ,v = iq and v = oo , and omitting water because of the practical 
nullity of the Adamantine Property for it proved by Rayleigh, 
we have positive values of tan a for six substances, from which, 
for these substances, the following Table of values of q/u ; and 
of tan" 1 fj, I, the differences of the Principal Incidence from the 
Brewsterian angle; has been calculated according to (123) and 
(118). The smallness of these last mentioned differences is very 
remarkable. 



Substance 


ff = qlu 


tan" 1 fj. - 1 


tau~ ] fj. 


"Sulfure d'arsenic trans- 
parent " (llealgar) 


01277 


0-08873 


67 '83 


" Blende transparente " 
(Zinc sulphide) 

Diamond 


00706 
00293 


0-04289 
0-00883 


67-13 
67-67 


Flint Glass 


00936 


0-00593 


59-89 


"Verre" 


00553 


0-0004% 


56-08 


Absolute Alcohol 


00284 


0-0000465 


53-13 



183. Another great fundamental province of Optics, luminous 
waves travelling through transparent crystals, was successfully 
explored by Fresnel more than eighty years ago. This he did 
with utterly imperfect dynamical leadings ; but nevertheless he 
discovered what we now know to be, in every detail except his 
equivoluminal condition, the true laws of light-waves in a crystal. 

184. One notable detail of Fresnel's, which I described in 
my introductory lecture (Oct. 1st, 1884), was that he made the 
propagational velocity of light in a crystal depend on the direction 
of the vibration, and not on the axis of the shearing rotational 



HUYGHENS CONFIRMED EXPERIMENTALLY BY STOKES. 425 

strain as the elastic solid theory has seemed to require. See Molar. 
Lecture I., pages 17, 18, particularly the words "If the effect 
" depends upon the return force in an elastic solid." In the same 
lecture (page 19) I told of an explanation of this difficulty 
suggested first by Rankine and afterwards by Stokes and by 
Rayleigh, to the effect that the different propagational velocities 
of light in different directions through a crystal are due, not to 
aeolotropy of elastic action, but to aeolotropy of effective inertia. 
But I had also to say that Stokes working on this idea, which 
had occurred to himself independently of Rankine's suggestion, 
had been compelled to abandon it for the reasons stated in the 
following reproduction of a short paper of twenty-one lines which 
appeared in the Philosophical Magazine for October, 1872 ; and 
which is all that he published on the subject : 

"It is now some years since I carried out, in the case of Iceland 
" spar, the method of examination of the law of refraction which 
" I described in my report on Double Refraction, published in the 
"Report of the British Association for the year 1862, p. 272. 
"A prism, approximately right-angled isosceles, was cut in such 
" a direction as to admit of scrutiny, across the two acute angles, 
" in directions of the wave-normal within the crystal comprising 
" respectively inclinations of 90 and 45 to the axis. The directions 
" of the cut faces were referred by reflection to the cleavage- 
" planes, and thereby to the axis. The light observed was the 
" bright D of a soda-flame. 

" The result obtained was, that Huyghens' construction gives 
" the true law of double refraction within the limits of errors of 
" observation. The error, if any, could hardly exceed a unit in 
" the fourth place of decimals of the index or reciprocal of the 
"wave-velocity, the velocity in air being taken as unity. This 
" result is sufficient absolutely to disprove the law resulting from 
" the theory which makes double refraction depend on a difference 
"of inertia in different directions. 

" I intend to present to the Royal Society a detailed account 
"of the observations; but in the meantime the publication of 
" this preliminary notice of the result obtained may possibly be 
" useful to those engaged in the theory of double refraction," 

It is well that the essence of the result of this very important 
experimental investigation was published : it is sad that we have 



426 LECTURE XIX. 

Molar, not the Author's intended communication to the Royal Society 
describing his work. 

The corresponding experimental test for a Biaxal Crystal, 
resulting also in a minutely accurate verification of Fresnel's 
wave-surface, was carried out a few years later by Glazebrook*. 
Must we therefore give up all idea of explaining the different 
velocities of light in different directions through a crystal by 
seolotropic inertia ? Yes, certainly, if, as assumed by Green and 
Stokes, ether in a crystal is incompressible. 

185. But Glazebrook has pointed out as a consequence of 
my suggestion of approximately zero velocity of condensational - 
rarefactional waves in a transparent solid or liquid, that with this 
assumption aeolotropic inertia gives precisely Fresnel's shape of 
wave-surface and Fresnel's dependence of velocity on direction 
of vibration, irrespectively of the direction of the strain-axis. 
Thus after all we have a dynamical explanation of Fresnel's laws 
of light in a crystal which we may accept as in all probability 
absolutely true. To prove this let us first investigate the con- 
ditions for a plane wave in an isotropic elastic solid with any 
given values for its two moduluses of elasticity ; k bulk-modulus 
arid n rigidity-modulus ; and with seolotropic inertia in respect to 
the motion of any small part of its substance. This seolotropy of 
effective inertia of ether through the substance of a transparent 
crystal follows naturally, we may almost say inevitably, from 
the Molecular Theory of 162168. In Lecture XX. details 
on which we need not at present enter will be carefully con- 
sidered. 

186. Meantime we may simply assume that Bp x , Bp y , Bp z 
are the virtual masses, or inertia-equivalents, relatively to motions 
parallel to x, y, z, of ether within a very small volume B containing 
a large number of the ponderable atoms concerned : so that 

' 



are the forces which must act upon the ether to produce com- 
ponent accelerations dgfdt, etc. Hence the equations of molar 

* "An experimental determination of the Values of the Velocities of Normal 
Propagation of Plane Waves in different directions in a Biaxal Crystal, and a Com- 
parison of the Results with theory." By R. T. Glazebrook, Communicated by 
J. Clerk Maxwell. Phil. Trans. Roy. Soc., 1879, Vol. 170. 



1 "I < .TROPIC INERTIA FOR WAVE-MOTION IN A CRYSTAL. 427 



motions of the ether will be (8) of 113, modified by substituting Molar. 
for p in their first members p x , p y , p z respectively. 

187. To express a regular train of plane waves, take 



(125), 

\\hrre X, /*, v denote the direction cosines of a perpendicular to the 
wave-planes ; and a, /3, 7 the direction cosines of the lines of 
vibration. These, used in (7) of 113, give 

where cos ^ = X -I- /* -I- yi> (127). 

Thus ^ is the inclination of the direction of the displacement, to 
the wave-normal. 

Using (124), (125), (126) in the equations of motion, and 
removing from each side of each the factor 

we find 

. X cos ^ a \ 
p z o.= (k+ Jn) l-w ; I 

i, rns \ /Q 

(129). 



-(*+*) 



187'. From these equations we determine the direction- 
cosines (a, ,7) of the vibration; and the propagational velocity, u, 
of the plane wave (X, /*, v) ; thus : First solving for o, /9, 7, and 
putting 

Px =n/a>, Py = njb\ p, = n/* (129'), 



428 LECTURE XIX. 

Molar, we find 



(130), 




hence by 



Multiplying the first of (130) by A,, the second by //,, the third by 
v and adding ; and removing the common factor cos ^ ; we find 



This is a cubic for determining u 2 , the square of the propagational 
velocity. The three roots are all obviously real ; one greater than 
the greatest of a-, ft 2 , c 2 , and the other two between the values of 
these quantities. For each value of u- the corresponding direction 
of vibration is given by (130). 

187". Calling uf, iiJ two of the three roots of (132) we find, 
by writing down the equation for each of these and subtracting 
one equation from the other, the following 



k + $n I" Vo_ 

n [_(ui 2 - a 2 ) - a 2 ) (, - 



6 2 ) (w 2 2 - 6-) 



And taking the corresponding notation in (130) we find 
k + 



\ cos ^ cos ^ ......... (132"). 

8 



H * 
(i 2 - c 2 ) (u* - c 8 ) 

Comparing these two equations, we see that 
cannot generally be zero : that is to say the vibrational lines 
corresponding to parallel wave-planes travelling with different 
velocities cannot generally be perpendicular to one another. This 



CONDENSATIONAL-RAREFACTIONAL WAVE VERY SLOW. 429 

result for the new theory, in which differences of velocity are due Molar. 
to inertial aeolotropy, presents an interesting contrast to the 
theorem of Lecture XII., page 136, that the vibrational lines in 
any two of the three waves are mutually perpendicular; when there 
is no inertial aeolotropy, and the differences of velocity are due to 
seolotropy of elasticity. 

188. In each of the two extreme cases of k = oo and k = f n 
the cubic sinks to a quadratic, as we most readily* see by (in 
virtue of X s + p? + i/ 1 = 1) writing (132) thus 

XV i'u* vW 




-6 u'-c 1 



189. From this we see that if k + $n is very great, one root 
of the cubic for w f is very great, being given approximately by 



(134); 

while the other two roots are approximately the roots of the 
quadratic, 

^ + 4V + J~ ............... (135). 

w* a* t* f 6 1 w* c* 

This' agrees with the propagational velocities for a given wave- 
plane (X, fj., v), found by Stokes and by Rayleigh from Rankine's 
hypothesis of aeolotropic inertia, and Green's assumption of a 
virtually infinite resistance against compression. It implies a 
wave-surface proved observationally by Stokes for Iceland spar 
(uniaxal), and by Glazebrook for arragonite (biaxal), to differ from 
the truth by far greater differences than could be accounted for 
by errors of observation. 

190. But ifk + $n is very small, one of the three values of u* 
given by (133) is very small positive, being given approximately 



* Another way of managing this detail will be found in the investigation of 
chiral waves in 20f> of Lee. XX. 



430 LECTURE XIX. 

Molar, while the other two are approximately the roots of the quadratic 

=0 ............ < 137 >- 



This is Fresnel's equation [see (94) of Lecture XV.] for the 
two propagational velocities for a given direction of wave-plane 
(X, n, v). It implies precisely Fresnel's celebrated wave-surface 



or r z (aW- + fry* + c 2 * 2 ) - ft 2 (6 2 + c 2 ) x- - 1)- (c 2 + ft 2 ) f - c 2 (a 2 + 6 2 ) 2- 



This equation is got, not now " after a very troublesome algebraic 
process," as said by Airy* in 1831, but by a very short and easy 
symmetrical method given by Archibald Smith in 1835-f; the 
problem being to find the envelope of all the planes given by the 
equation 

\oc + fiy + vz = u \ 

with X a + /* 2 + v a =ll .................. (138"). 

and (137) for u ) 

191. The theorems of Glazebrook and Basset, stated in 
195 below, are readily proved by using (139) in connection with 
Archibald Smith's now well-known investigation of Fresnel's wave- 
surface. But the theory on which it is founded implies essen- 
tially condensation and rarefaction and therefore a direction of 
vibration not in the wave-plane ; and not agreeing with Fresnel's 
which is exactly in the wave-plane (corresponding as it does to 
strictly equivoluminal waves). For our present case of k + $n = 0, 
the vibrational direction (a, /3, 7) as given by (130) is 

2 X cos ^ b-a cos ^ c-v cos ^ 

a = , p ,,, 7 = ...... (l.iU); 

u- a- u- o- u- c 2 

and (131) for ^V, the angle between the vibrational direction and 
the wave-normal, becomes 



, 
u 2 -a- \u--b- 

Cannot g i\f at j ieinatica i Tracts, p. 353. 

COrreSpond/amftrufye Phil. Soc. Vol. vi. p. 85; also Phil. Mag. Vol. xn. 1838 (1st 

velocities ca'1 5 - 



VIBRATIONAL LINES NOT EXACTLY IN WAVE-PLANE. 431 

192. For an example, take arragonite with, as iu Lee. XV. Molar. 
37, (and n = 1 for simplicity) 



1/6 = 16816; b = '59467; 6* = '35363 
1/c = 1'6859 ; c = '59315 ; c' = '35183 



... (141); 



and let us find the two propagational velocities (ti,, it?) and the 
inclinations (^,, ^,) of the vibrational lines to the wave-normal, for 
the case in which the wave-normal is equally inclined to the three 
principal axes (X = n = v = l/x/3). By the solution of the quadratic 
(137) and by using the roots in (139) we find 

,' = -40234; r' = l-57653; ^ = 8137''68 ( 
< = -35272; t/ a -' = 1 '68377 ; * 2 = 8949'-24 )'" 

Remarking that w," 1 and w," 1 are the indices of refraction for the 
two waves of which the normals are each equally inclined to the 
three principal axes (x, y, z) it is interesting to notice how nearly 
the greater of them is equal to (1 '6816 + 1 '6859), the mean of the 
two greater of the three principal refractive indices. This, and the 
nearness of ^, to 90, are due to the smallness of the difference 
between the two greater principal indices: that is to say, the 
smallness of the difference between arragonite and a uniaxal 
crystal. In fact w^ 2 , the second of our solutions of the quadratic, 
corresponds to what would be the ordinary ray if the two greater 
principal indices were equal. 

193. To help in thoroughly understanding the condensations 
and rarefactions which the theory gives us in any plane wave 
through a biaxal crystal, take as wave-plane any plane parallel to 
one of three principal axes, OY for instance. It is clear without 
algebra that the directions of vibration in the two waves for 
every such direction of wave-plane are respectively parallel and 
perpendicular to OY. The former corresponds to the ordinary 
ray: its vibratioual direction is parallel to OF: it has propa- 
gational velocity */(n{p y ), or 6, according to (129'). It is a strictly 
equivoluminal wave. All this we verify readily in our algebra by 
putting /*=0; which gives for one root of the quadratic (135) 
w a = b-, and gives by (139) see 8 ^ = 00 and therefore ^ = 90. 



432 LECTURE XIX. 

Molar. 194. For the other root of the quadratic (135) we have 

V _*_ 
u*-a 2 tf-c* 

whence u 2 = c 2 X 2 + aV ; 

tt 8 -a" = (c 8 -a")\ 8 ; (143). 

Using these in (139) we find 



a a 

Putting now y = in (138'), the equation of Fresnel's wave- 
surface, we find for its intersection with the plane XOZ 

(r 2 -& 2 )(aV + cV-V) = ............... (145). 

This expresses, for the ordinary ray, a circle r = b ; and for the 
extraordinary ray an ellipse, 



The wave-plane touches this ellipse at the point (x, z)\ hence 
* = '?.. ...(147). 

V CZ 

Talcing this for \jv in (144) we see that the vibrational line is 
perpendicular to the ray-direction, which is the radius vector of 
the ellipse through the point (x, z). This is only a particular case 
of the general theorem given by Glazebrook, Phil. Mag. Dec. 1888 
page 528, that the direction of vibration is perpendicular to the 
ray in the new theory of double refraction founded on aBolotropic 
inertia. 

195. Moreover in this theory* Basset has given a most 
interesting theorem f to the effect that the direction of the vibration 
is a line drawn perpendicular to the radius vector from the foot 
of the perpendicular to any plane touching the wave-surface (the 
perpendicular to the radius vector being drawn from the centre of 

* Basset's Physical Optics (Cambridge, 1892), 265. 

t Given also for a very different dynamical theory involving zero velocity of con- 
densational-rarefactional wave, by Sarrau in his Second Paper on the Propagation 
and the Polarization of Light in Crystals ; Liouville's Journal, Vol. xm. 1868, p. 86. 



GLAZEBROOK AND BASSET'S DRAWING FOR VIBRATIONAL LINE. 433 

the wave-surface to the tangent-plane and to its point of contact). Molar. 
This includes Glazebrook's theorem and appends to it a simpler 
completion of the problem of drawing the vibrational line than 
that given by Glazebrook in pages 529, 530 of the volume of the 
Philosophical Magazine already referred to. The construction is 
illustrated in figure 13, drawn exactly to scale for the principal 
section through greatest and least principal diameters of wave- 
surface for arragonite. P is the point of contact of the tangent- 
plane KM. OP is the radius vector (optically the ray). F is the 
foot of the perpendicular from the centre of the wave-surface. 
FN, perpendicular to OP, not shown in the diagram, is the direc- 
tion of the vibration. It would be interesting to construct on 
a tenfold scale a portion of figure 13 around FP; but it is perhaps 
more instructive to calculate the angle NFP (which is equal to 
FOP) and the radius of curvature, at P, of the ellipse. Figure 13 
illustrates the construction for any wave-plane whatever touching 
the wave-surface in the point P ; though it is drawn exactly to 
scale only for the extraordinary ray in the principal section of 
arragonite, through the greatest and least principal diameters. 




Fig. 13. 



T. L. 



28 



434 . LECTURE XIX. 

Molar. 196 Going back now to 194 we see that the intersections 
of Fresnel's wave-surface with the three planes YOZ, ZOX, XOY, 
are expressed respectively by the following equations 

- 6 2 c 2 ) = \ 

- c 2 a 2 ) = I (148), 

(r 2 - c 2 ) (aV + &y - 6 2 a 2 ) = J 

which prove that each intersection consists of an ellipse and a 
circle ; the ellipse corresponding to the extraordinary ray, and the 
circle to the ordinary. 

197. Lastly consider wave-planes perpendicular to one or 
other of OX, OY, OZ. Take for example OX ; we see that the 
two propagational velocities are b and c, with vibrational lines 
respectively parallel to OF and OZ. The physical explanation 
is much more easily understood than anything which we thought 
of in Lecture I. pages 17 to 20, when we were believing that 
differences of velocity were due to seolotropy of elasticity. We 
now see that the two waves travelling #-wards have different 
velocities because of greater or less effective inertias of the moving 
ether in its vibrations parallel respectively to Y and OZ. 

198. The fundamental view given by Fresnel for the 
.determination of his wave-surface by considering an infinite 
number of wave-planes in all directions through one point, and 
waves starting from them all at the same instant, is most im- 
portant and interesting ; truly an admirable work of genius ! 
It leaves something very definite to be desired in respect to the 
geometry and the dynamics of a real source of light travelling 
in all directions from a small portion of space in which the source 
does its work. It therefore naturally occurs to consider what may 
be the very simplest ideal element of a source of light. 

199. Preparation was made for this in the "molar" divisions 
of Lectures III.... VI. and VIII.... XIV. and particularly in pages 
190 to 219 of the addition to Lecture XIV. What we now want 
is an investigation of the motion of ether in a crystal due to an 
ideal molecular vibrator moving to and fro in a straight line in 
any direction. The problem is simplified by supposing the direc- 
tion to be one of the three lines OX, OY, OZ, of minimum, and of 
minimax, and of maximum effective inertia of ether in the crystal. 



THEORY FOR VIBRATIONAL AMPLITUDE IN WAVE-SURFACE. 435 

It seems to me that this should be found to be a practicable Molar, 
problem. Towards its solution we have Fresnel's wave-surface 
as the isophasal surface of the outward travelling disturbance or 
wave-motion : and we have the direction of the vibration in each 
part of the surface by the theorems of Glazebrook and Basset 
( 195 above). What remains to be found in our present problem 
is the amplitude of the vibration at any point of the wave-surface. 
One thing we see without calculation is, that at distances from 
the origin great in comparison with the greatest diameter of the 
source, the vibrational amplitude is zero at the four points in 
which the wave-surface is cut by the vibrational line produced 
in both directions from the source ; when this line coincides with 
one of the three axes of symmetry OX, OY, OZ. The general 
solution of the problem is to be had by mere superposition 
of motions from the solutions for vibrations of the source in the 
three principal directions. For the present I must regretfully 
leave the problem hoping to be able to return to it later. 



282 



LECTUEE XX. 

FRIDAY, October 17, 5 P.M., 1884. Written afresh, 1903. 

Molecular. 200. CONSIDERING how well Rankine's old idea of aeolotropic 
inertia has served us for the theory of double refraction, it natur- 
ally occurs to try if we can found on it also a thorough dynamical 
explanation of the rotation of the plane of polarization of light in 
a transparent liquid, or crystal, possessing the chiral property. 
I prepared the way for working out this idea in a short paper 
communicated to the Royal Society of Edinburgh in Session 
187071 under the title "On the Motion of Free Solids through 
a Liquid " which was re-published in the Philosophical Magazine 
for November 1871 as part of an article entitled " Hydrokinetic 
Solutions and Observations," and which constitutes the greater 
part of Appendix G of the present volume. The extreme diffi- 
culty of seeing how atoms or molecules embedded in (ether), an 
elastic solid could experience resistance to change of motion 
practically analogous to the quasi-inertia conferred on a solid 
moving through an incompressible liquid has, until a few weeks 
ago, prevented me from attempting to explain chiral polarization 
of light by oeolotropic inertia. Now, the explanation is rendered 
easy and natural by the hypothesis explained in 162 164 above 
and in 204, 205 below and in Appendix A. 

201. To explain aeolotropic inertia, whether chiral or not, of 
molecules in ether, from the rudimentary statements in Appendix A, 
take first the very simplest case; a diatomic molecule (A Jt A 2 ) con- 
sisting of two equal and similar atoms held together by powerful 
attraction ; so as, with a single electrion in each, to constitute a 
rigid system when, as we shall at first suppose, the forces and 
motions with which we are concerned are so small that the 



ELECTRO-ETHERIAL EXPLANATION OF .fiOLOTROl'lC INERTIA. 437 

electrions have only negligible motion relatively to the atoms. Molecular. 
This supposition will be definitely modified when we come, in 
I 242, to explain chromatic dispersion on the new theory. 



202. In fig. 14 the circles represent the bounding spherical 
surfaces of the two atoms. According to the details suggested for 
the sake of definiteness in Appendix A and illustrated by its 
diagram of stream-lines (fig. 5), the two atoms must overlap as 
indicated in our present diagram fig. 14, if the stream-lines of 
ether through each atom are disturbed by the presence of the 
other. Without attempting any definite solution of the extremely 
difficult problem of determining stream-lines of ether through our 
double atom we may be at present contented to know that the 
quasi-inertias of the disturbed motion of ether within the molecule, 
in the two cases in which the motion of the ether outside is parallel 
to A^Ai, and is perpendicular to A^ t must be different. It seems 




Fig. 14. 

to me probable that the former must be less than the latter: I shall 
only assume however that they are different, which is certainly 
true: and I denote the former by a and the latter by fi. This 
means that if the molecule is at rest, and the ether outside it is 
moving uniformly according to velocity-components (f, 17) respec- 
tively parallel to A l A< t , and perpendicular to it in the plane of 
the diagram, the kinetic energy of the whole etherial motion will 
be greater by ^(oi 2 + /3?7 2 ) than if the ether had the uniform 
motion (, 17) everywhere. Thus if B denote any volume of space 



438 LECTURE XX. 

Molecular, completely surrounding one molecule but not including another, 
and if p be the undisturbed density of ether, the kinetic energy of 
etherial motion within B is 



Throughout 202 205 we are supposing the different constituent 
molecules of the assemblage to occupy separate spaces. In 225 
we shall find ourselves obliged to assume overlapping complex 
molecules of silica in a quartz crystal. 

203. This gives a clear and definite explanation of the 
seolotropy of inertia ( 184 above) suggested by Rankine for 
explaining double refraction. First for a uniaxal crystal, consider 
any homogeneous assemblage (Appendix H, 3, 6, 15, 16, 17, 18, 
19) of our diatomic molecules and let B be the volume of space 
allotted to each of them. The homogeneousness of the assemblage 
implies that the lines A 1 A 2 in all the molecules are parallel. Take 
this direction for OX; and for OF, OZ any two lines perpendicular 
to it and to one another. In respect to double refraction it is of 
no consequence what the character of the homogeneous assemblage 
may be : though it may be expected to be symmetrical relatively 
to the direction of A l A y because the equilibrium of the assemblage, 
and the forces of elasticity called into play by deformation, depend 
on mutual forces between the molecules not directly concerned 
with the elasticity and motions of the ether called into play in 
luminous waves. In short, the essence of our assumption is that 
the molecules are unmoved, while the ether is moved, by waves of 
light. Write (149) as follows 



p + + P + 

where 

a 8 

px=P + g; py = P + ^ ............... (151). 

This shows clearly the meaning, and the physical explanation, 
of the p x , p y , p z assumed in 186 as virtual densities of ether 
relatively to motions in the three rectangular directions. With 
diatomic molecules having their axes parallel to OZ, we have 
essentially p y = p x . This gives the optical properties of a uniaxal 
crystal, which essentially present no difference between different 
directions perpendicular to the axis; though the homogeneous 



DYNAMICAL EXPLANATION OF A CHIRAL MOLECULE. 439 

assemblage of molecules, constituting the crystal, presents es- Molecular. 
sentially the differences in different directions corresponding to 
tactics of square order, or of equilateral-triangle order. 

204. When instead of the mere diatomic molecule of 201 we 
have in each molecule a molecular structure which is isotropically 
symmetrical (square order or equilateral-triangle order) in planes 
perpendicular to one line OZ, we still have p v = p x , giving the 
axially isotropic optical properties of a uniaxal crystal : and this 
quite independently of any symmetry of the homogeneous assem- 
blage of molecules constituting the crystal, if the virtual inertia 
contributed to the ether by each molecule is independent of its 
neighbours. If the structure of each molecule has no chirality 
(Appendix H, 22, footnote) the homogeneous assemblage has no 
chirality. And if each molecule is geometrically and dynamically 
symmetrical with reference to three rectangular axes (OX, OY, OZ), 
but not isotropic in respect to these axes, we have generally three 
different values for p x , p y , p z ; and ( 187, 190 above) exactly 
Fresnel's wave-surface. This is quite independent of any sym- 
metry of the homogeneous assemblage constituting the crystal : it 
is merely because the axes of symmetry of all the molecules are 
parallel in virtue of the assemblage being homogeneous. For 
brevity I now call a molecule which has chirality, a chiroid. 

205. When each molecule is a chiroid it may (Appendix G, 
Part 1) contribute a chiral property to the inertia of ether oscil- 
lating to and fro in the space occupied by the assemblage. To 
understand this chiral inertia, consider a volume B of ether, very 
small in all its diameters in comparison with a wave-length of 
light, and exactly equal in volume to the volume of space allotted 
(201) to each molecule. Let (, 77, ) and (, 77, ) be the components 
of displacements and velocity of ether within B but not within 
the part or parts of B occupied by the atoms of the molecule. 
The components round x, y, z, of rotational velocity (commonly, 
but perhaps less conveniently, called angular velocity) of ether in 
space in the neighbourhood of B and not within any atom, are 



_ _ 

2dy dz' 2\dz dx' 2 ? dy 

Let x*> Xv> Xt ^ e coefficients which we may call the inertial 
chiralities of the molecule relative to x, y, z respectively. The 



440 LECTURE XX. 

Molecular, chiral inertia with which we are concerned may be denned by 
asserting that acceleration of any one of the three components 
(152) of angular velocity of the ether, implies a mutual force 
between the molecule and the ether, in the direction of the axis 
of the rotational acceleration, which may be expressed by the 
following equations* 



. n- -_ 7?= _ 

Xx ~dt\dy dz)' ^~ Xy dV\dz dx)' * z dt\dx "dy 

......... (153); 

where P, Q, R, denote components of force per unit of volume, 
exerted by the molecules on the moving ether. Hence the 
x, y, ^-components of the elastic force on ether per unit of its 
volume acting against inertial reaction are equal to 



The equations of motion of ether occupying the same space as 
our homogeneous assemblage of molecules will be found by substi- 
tuting (154) for the first members of (8) in 113. Thus with (153) 
we find 

d*% d* (dS d<n\ ., .dfdd'ndS\ l n ., fc 

Px -T5 - Xx ~TTo (-T- TT ( + VO j- j 5 + j + j^ + " C 

r* fa A*dt z \dy dz) dx\dx dy dzj 

......... (155), 

and the symmctricals in relation to y and z\ three equations to 
determine the three unknowns , 77, ^. 

206. The method of treatment by an arbitrary function, as 
in 115, 120, 121, 127, 130, would be interesting; but because of 
the triple differentiations in the chiral terms it is not convenient. 
All that it can give is in reality, in virtue of Fourier's theorems, 



* The negative sign is prefixed to x i these equations in order to make x 
positive for a medium in which right-handed circularly polarized light travels faster 
than left-handed (see 215 below). Such a medium is by all writers on the subject 
called a right-handed medium, because the vibrational line of plane polarized light 
travelling through it turns clockwise as seen by a person testing it with a Nicol's 
prism next his eye. The molecule which, according to our inertial theory, produces 
this result is analogous to a left-handed screw-propeller in water and is therefore 
properly to be called left-handed. Thus left-handed molecules produce an optically 
right-handed medium. 



FORMULAS EXPRESSING CHIRAL INERTIA IN WAVE-MOTION. 441 

included in the more convenient method of periodic functions ; of Molar. 
which the most convenient form for our present problem is given 
by the use of imaginaries with the assumption , 17, f equal to 
constants multiplied into 

(t **+w+**\ 
i~t ' ....................... (156), 

where X, yn, v denote the direction-cosines of a perpendicular to 
the wave-plane, and u the propagational velocity. This gives 

d _ ia>\ d _ idJfA d _ IQJV _, _ G>' d 1 _ s 

fa --- ^~ ; djj~ ~^T ; ^~'1T ; "'' dp- 

......... (157), 

which reduces (155) and its symmetrical to 



...... (158)*.' 

Multiplying each member of these equations by w 2 o>~ s and arrang- 
ing in order of , 17, f, we find 

(A - Ar'X 5 ) + (i-XfUav - #V) i? + (- iXxUa>n - k'v\) % = \ 

= L.(159), 

j 
where 

^ = ^ w s - ?i ; 5 = ^M' - n ; (7 = p z w - n ; ^' = (k + |n). . .(160). 
Forming the determinant for elimination of the ratios , ;, f ; and 
simplifying as much as possible, with reductions involving 

X a + fi + i/=l ...................... (161); 

and putting 



(x* - 



These formulas, implying as they do that the chirality expressed by them is 
due, not to chirality of elasticity but to chirality of virtual inertia, are given by 
Boussinesq on p. 456 of Vol. n. (1903) of his Thtorie Analytique de la Chaleur. 



442 LECTURE XX. 

Molar, we find 



** ( A - *0 + XzXxH* (B - k') + XxXy S~ (C-k')] 
Q ....................................... (163). 



In (163) we have an equation of the sixth degree for the 
determination of u, the propagational velocity of waves whose 
wave-normal has X, /*, v for its direction-cosines. 

This equation is greatly simplified by our assumption of 
practically zero propagational velocity of condensational-rare- 
faciional waves; which makes k' = n, and therefore by (160), 
and (129') 

, X 2 _ a 2 X 2 uW 

- ~T - ~9 9 - ~9 - 9 - ""'' 

A u 2 a 2 v? a? 
This, with corresponding formulas for /j?/B and v*/C, gives 



By k r = n, we also have 

A k' = p x u~ ; B k' = p y u- ; C k' = p 2 u". 
Thus (163) is reduced to 



. 



= 0...(163'). 

207. The imaginary term in (163), unless it vanishes, makes 
every one of the six values of u imaginary. The realisation of the 
corresponding result according to the principles of 150, 151, 
above will be very interesting. It essentially demands an ex- 
tension of the dynamical theory to include the conversion of the 
energy of wave-motion into thermal energy, energy of irregular 
interrnolecular motions ; that is to say the dynamics of the 
absorption of luminous waves travelling through an assemblage 
of atoms or of groups of atoms. 

208. Remark first that when E does not vanish, the 
imaginary term in (163) vanishes if, and only if, one of the three 



SPEED OF CIRCULARLY POLARIZED LIGHT IN CHIRAL MEDIUM. 443 

direction-cosines \, p, v, vanishes ; that is to say if the wave-plane Molar, 
is parallel to one of the three principal axes. It is certainly a 
curious result that plane waves cun be propagated without 
absorption if their plane is parallel to any one of the principal 
axes; while there is absorption for all waves not fulfilling this 
condition : i other words that the crystal should be perfectly 
transparent for all rays perpendicular to a principal axis, and 
somewhat absorptive for all rays not perpendicular to a principal 
axis. No such crystal is known in Nature or in chemical art. 
Time forbids us to go farther at present into this most interesting 
subject. 

209. Considering now cases in which E vanishes and there- 
fore (1C3) has all its terms real, we see that in all such cases it 
becomes a cubic in u\ A first and simplest case of this condition 
is indicated by the third member of (162), which shows that E 
vanishes when p x = p v = p z . For this case (163) becomes 
(pu'- n - 
where 

The three roots of this cubic are given, one of them by equating 
the first factor to zero, and the two others by the quadratic in u a 
obtained by equating the second factor to zero. No crystals are 
known to present the optical properties thus indicated for unequal 
values of % x , j^, %,. But it is conceivable that these properties 
may be found in some crystals of the cubic class, which might 
conceivably, while isotropic in respect to ordinary refraction, give 
different degrees of rotation of the plane of polarization of polarized 
rays travelling in different directions relatively to the three rect- 
angular lines of geometrical symmetry. 

210. For the case of x x = x v = % z the quadratic of (164) 
becomes (pit 2 ?i) 2 = cafyW whence 

pu* n = 1 w^Jt (165). 

In all known cases, I think we may safely say in all conceivable 
cases, whether of chiral crystals, or of chiral liquids, w^u is very 
small in comparison with n; in other words (213 below) the 
difference between the propagational velocities of right-handed 
and left-handed circularly polarized light is exceedingly small 
in comparison with the mean of the two velocities. Hence we 
lose practically nothing of accuracy by taking u = </(n/p) in the 



444 LECTURE XX. 

Molar, second member of (165). Thus if we denote by u lt w 2 the two 
positive values of u given by (165) and by u a mean of the two, 
we find by (165) 



N ..... (166). 



u' 2 n ' u n pu 

And, in accordance with (182), putting 



we have 

u i Uz =r g .......................... (166"). 

211. To interpret this result go back to (158) simplified by 
making p x = p u = p z and ^ = ^ = ^. The medium being isotropic 
in respect to all directions of the wave-plane, we lose nothing of 
generality by putting X = l, yu, = 0, v = 0. With these simplifi- 
cations, the first of the three equations (158) gives =0 and the 
second and third multiplied by w 2 / * 3 become 

(pit 2 n)rj = r^eow^; (pu- n) = lywui) ...... (167). 

Equating the product of the first members to the product of the 
second members of these equations we find (pu- n)- = % 2 a> 2 M 2 , 
which verifies the determinantal quadratic as given at the 
beginning of 210. 

Using (165) to eliminate (pif ii) from (167) we find 

?=7 ........................... (168). 

Hence as an imaginary solution according to (156) of 206 we 
have for , 77, respectively 



0; tfe; ^C ............. (169). 

Changing the sign of i gives another imaginary solution ; and 
taking the half-sum of the two imaginary solutions, we find as 
a real solution 



= 0; 9i = CcQa<ot--, = + Csinw - 





212. The interpretation of this is, sinusoidal vibrations of 
equal amplitudes, parallel respectively to Y and OZ ; the former 



COEXISTENT WAVES OF CONTRARY CIRCULAR MOTIONS. 445 

being a quarter of a period behind or before the latter, according Molar 
as we choose the upper or the lower of the two signs in each 
formula. The resultant (y, z) motion of the ether*, in the case 






Fig. 15. 




* This (y, ) component is irrotational. The (x, y) component of the etherial 
motion is rotational having OZ as axis ; and the (x, z) component is rotational having 
OY as axis. We are not concerned with this view at present. But we meet it 
necessarily when we think out the geometry of 205 (152) ; and it brings to us a 
very simple synthetic investigation of the velocity of circularly polarized light in a 
chiral liquid or in an isotropic chiral solid. 



446 LECTURE XX. 

Molar, represented by the lower signs, is in circular orbits in the direction 
shown as anti-clockwise in the annexed diagram (fig. 15). This 
motion is the same in phase for all points of the ether in every 
plane perpendicular to OX, that is to say every wave-plane ; and 
it varies from wave-plane to wave-plane so as to constitute a wave 
of circularly polarized light travelling #-wise with velocity u. The 
motion corresponding to the upper signs is another circularly 
polarized wave with opposite orbital motion, that is to say clock- 
wise direction. The radius of the orbit is the same, C, in the two 
cases. If* when looking to fig. 15, we take the positive OX as 
towards the eye, a line of particles of the ether which is parallel to 
OX when undisturbed, becomes, in the wave of clockwise orbits, 
a right-handed spiral ; and in the wave of anti-clockwise orbits a 
left-handed spiral. Thus the two waves are of opposite chiralities : 
the former are called right-handed, the latter left-handed. The 
steps of the two spirals (screws) are slightly different, being the 
spaces travelled by the two waves in their common period; that 
is to say the wave-lengths of the two waves. To understand this 
look at fig. 16 showing a right-handed spiral RRR of step 6 cm. 
and a left-handed spiral LLL of step 5 cm. ; having a common 
axis OX, and both wound on a cylinder of radius 1^ cm. repre- 
senting C of (170). If the radius of this cylinder were reduced to 
a thousandth or a millionth of the step of either screw, and if the 
step were reduced to about 4. 10~ 5 cm., either spiral might represent 
the line in which particles of ether lying in OX when undisturbed 
are displaced by homogeneous yellow or yellow-green circularly 
polarized light, travelling in the direction of OX positive, through 
a transparent liquid or solid of refractive index of about 1'5. With 
this understanding as to the scale of the diagram consider the 
resultant of the two equal coexisting displacements of wave-planes 
represented at any instant by the two spirals. At points of inter- 

* This is a convention which I have uniformly followed for sixty years in 
respect to the positive and negative directions in three mutually perpendicular lines 
OX, OY, OZ. It makes the positive direction for angular velocities be from OX to 
OY, from OY to OZ, and from OZ to OX. This agrees with the ordinary conven- 
tions of English as well as foreign books on trigonometry, and geometry of two 
dimensions, which make anti-clockwise rotation be positive. It is convenient for 
inhabitants of the Northern Hemisphere as it makes positive the anti-clockwise 
orbital and rotational motions of the sun and planets as viewed from space above 
our North Pole, which we may call the north side of the mean plane of the 
planetary motions. 



FRESNEL'S KINEMATICS OF COEXISTENT WAVES. 447 

section of the two spirals, for example D, D', D", the resultant Molar. 
displacement of each of the corresponding wave-planes is 2(7, the 
sum of the two equal components. The resultant displacement 
of the wave-plane through AB is twice the distance from OX of 
the middle point of AB, that is to say 2 *J(C*~- \AB*). 

213. Suppose now the two spirals to rotate in opposite 
directions, with equal angular velocity o>, round the axis OX; 
the right-handed spiral clockwise and the left-handed anti-clock- 
wise when viewed from X towards 0. This may be realised in an 
instructive model having one spiral wound on a brass or wooden 
cylinder, and the other on a glass tube fitting easily around it. 
The two moving spirals will represent the motions of the ether in 
two circularly polarized waves travelling awards with velocities 



X lt X-j denoting the steps of the two screws. The motion of the 
ether in any fixed plane perpendicular to OX, the plane through 
AB for instance in the diagram, will be found by the geometrical 
construction of 212. Thus we see that a very short time after 
the time of the configuration shown in the diagram, the point D 
will come to the plane through AB, and the points A, B will 
come together at D: immediately after this they will separate, 
A leftwards in the Diagram, B rightwards. Considering the whole 
movement we see that in any one fixed plane through OX, a 
wave-plane of the ether moves to and fro in a fixed straight line. 
The point D will travel #-wards with a velocity equal to 




being the harmonic mean of the velocities of the two compounded 
circular waves; and will revolve slowly clockwise round the 
cylinder of radius G at a rate, in radians per unit of distance 
travelled a;- wise, equal to 






Compare with (179) and (180) below. 



448 LECTURE XX. 

Molar. 214. The short algebraic expression and proof of all this is 
found by taking the two solutions represented by (170), (171) and 
writing down their sum with modification as follows : 

77 / X\ ( X\ 
-^ = COS 0) it j + COS & It J 



C 



X X \ O) [X X 

- -- ~) cos- ( --- ...... (175); 

2w 2 2uJ 2 V^2 MI 



--sin- ...... (176). 

2wo 2V 2 \9 uJ 

For magnitude and direction of the resultant of these we find 

V(,f+n = 2C'co S a>(*-^-^) ...... (177), 

and 

wfx x\ /i* \ 

- = tan = [ --- .................. (178). 

rj 2 \U, uJ 

These equations express rectilinear vibration through a total 
range 46' in period , in the line whose azimuth is 



This shows that the vibrational line in plane polarized light 
revolves clockwise in the wave-plane at a rate r, in turns per unit 
of space travelled, expressed by 



r - = '-- 2 = l l -* ...... (180). 

4-7T V< 2 W l/ 4<7r W 1 W 2 Ta>2 

In the last member T denotes the period of the vibrations, and a 
denotes Vw^, or the "geometric mean" of the propagational 
velocities, of the two waves. 

215. Returning now to figure 16 we see that, as virtually 
said in 21 2, 213; 

I. The orbital motions in right-handed circularly polarized 
light coming towards the eye are clockwise, and in left-handed 
anti- clockwise. 



LEFT-HANDED MOLECULES MAKE A RIGHT-HANDED MEDIUM. 449 

II. The vibrational line of plane polarized light traversing Molar, 
a chiral medium towards the eye, turns clockwise when the 
velocity of right-handed circularly polarized light is greater, and 
anti-clockwise when it is less, than that of left-handed circularly 
polarized light. 

III. A chiral medium is called optically right-handed or 
left-handed according as the propagational velocity of right-handed 
or of left-handed circularly polarized light travelling through it 
is the greater. 

All this ( 212 215) is Fresnel, pure and simple. It is his 
kinematics of the optical right-handed and left-handed chirality 
discovered by Arago and Biot in quartz, and by Biot in turpentine 
and in a vast number of other liquids. 

216. For the dynamical explanation take the third member 
of (180); and look back to (152) and (153) which show that, 
according to our notation, twice the acceleration of angular velocity 
of moving ether multiplied by ^ is, when ^ is negative, a force 
per unit bulk of the ether in the positive direction in the axis of 
the angular velocity, exerted on the ether by the fixed chiral 
molecules. If % is positive, the chiral molecule is, as said in the 
footnote on 205, analogous to a left-handed screw and its chirality 
is properly to be called left-handed : because ( 205 above) if the 
ether concerned is viewed in the direction of the axis of the 
angular velocity, the force of left-handed molecule on ether is 
toward the eye (or positive) when the acceleration of the angular 
velocity of the ether is clockwise (or negative according to my 
convention regarding direction of rotation, stated in the footnote 
on 212). With this statement the simple synthetic investigation 
of the velocity of circularly polarized light in a chiral medium, 
indicated in the first footnote of 212, is almost completed; and 
by completing it we easily arrive, by a short cufc> at the solution 
expressed in the third member of (180), for an isotropic medium ; 
without the more comprehensive analytical investigation of 205 
to 214. 

By (166) we saw that the left-handed molecule (% positive) 
gives the greater propagational velocity (#,) for right-handed 
circularly polarized light, and therefore ( 215, III.) the medium 
is optically right-handed. Thus our conventions necessarily result 
in left-handed molecules making a right-handed medium and 
right-handed molecules a left-handed medium. 

T. L. 29 



450 LECTURE XX. 

Molar. 217. Going back now to 206, 207, 208, for light traversing 
a chiral medium with three rectangular axes of symmetry cor- 
responding to maximum, minimax and minimum wave velocities, 
let us work out the solution for wave-plane perpendicular to one 
of the principal axes, OZ for example. This makes \ = 0, //, = 0, 
i/=l; and, with (160), reduces the determinantal equation (162) 
to 

(C - k') [( Px u* - n) ( Py n"~ - n) - uWxxXy] = ... (181). 
Hence, removing the first factor (which, when k' = - n, gives u = 0, 
for the condensational-rarefactional wave) and putting 



Px py Pxpy 

we find 

( W 2 -a 2 )(M 2 -& 2 ) = <fu 2 ............... (183); 

a quadratic of which the two roots are the squares of the velocities 
of the 2- ward waves. The third of equations (158) makes z = Q 
for each of these waves ; and therefore they are both exactly 
equivoluminal. When foXy * s positive the two roots of the quad- 
ratic are both positive ; one of them > a 2 , the other < 6 2 if a 2 > b' 2 . 

218. In these formulas a and b denote the velocities of 
light having its vibrational lines parallel to OX and OY respec- 
tively ; and g is a comparatively very small velocity measuring 
the chiral quality of the crystal. Judging by all that has been 
hitherto discovered from observation of chiro-optic properties of 
gases, liquids and solids, we may feel sure that g is in every case 
exceedingly small in comparison with either a or 6 : it is about 
one twenty-thousandth in quartz : and in cinnabar, which has the 
greatest optic chirality hitherto recorded for any liquid or solid so 
far as I know, g seems to be about fifteen times as great as in 
quartz. Suppose now that g~ is very small in comparison with 
the difference between a? and 6 2 , we see, by the form of (183), that 
the two values of u* given by the quadratic must be to a first 
approximation equal to a 2 and 6 2 respectively : hence to a second 
approximation we have (taking forms convenient for the case of 
a 2 >6 2 ), 



219. To find the character of the two waves corresponding 
to the two roots of (183), put \ = 0, /i, = 0, */=! in the first and 



OPTIC CHIRALITY IMPERCEPTIBLE IN A BIAXAL CRYSTAL. 451 

second of (158), each multiplied by tt'or*; which, with (182),.Molar. 
reduces them to 



Pv 

Equating the product of the first members to product of the 
second members of these equations we verify the determinantal 
quadratic (183). With either value of u> given by (183), either 
one or other of equations (185) may be used to complete the 
solution. The first of (185), however, is the more convenient for 
the root approximately equal to 6 1 ; and the second for the root 
approximately equal to a 2 . Taking them accordingly and realising 
in the usual manner, we find as follows for two completed in- 
dependent solutions with arbitrary constants C lt C, : 




(186). 



220. The exceeding smallness of g/a* has rendered fruitless 
all attempts hitherto made, so far as I know, to discover optic 
chirality in a " biaxal crystal," that is to say a crystal having 
three principal axes at right angles to one another of minimum, 
minimax and maximum wave -velocities. If it were discoverable 
at all, it certainly would be perceptible in light travelling along 
one of these three principal axes. 

[Dec. 19, 1903. I have only to-day seen in Phil. Mag., Oct. 
1901 that Dr H. C. Pocklington has found rotations, per cm., 
22 anti-clockwise, and 64 clockwise, of the vibrational line in 
polarized light travelling through crystallized sugar along two 
" axes," of which the first is nearly perpendicular to the cleavage 
plane : a mo$t interesting and important discovery.] 

221. For light travelling along the axis, OZ, of a uniaxal 
crystal, take a = b in (183) (186), in this case, and we have 
by (183) 

u*-a*=gu ...................... (187). 

* For sodium-light in quartz it is -4605 . IQr* ; and it may be expected to be 
correspondingly small in biaxal crystals. 

292 



452 LECTURE XX. 

Molar. And, taking p y = p x , and % y = % x , we have, by (182), 

g = ?*S ........................... (188). 

With this notation and with (187), (186) becomes 

u-i > a ; = C-L cos a>(t --- ); ?/ = - (7j sin &> ( t --- ) I 

uj V "A ...(188'). 

u 2 < a ; = (7 2 sin &> u ) ; T; = C, cos o> ( t -- ) 

\ U 2 ' \ U 2 / / 

Thus we see that for light travelling along the axis of a uniaxal 
crystal the velocities of right-handed and of left-handed circularly 
polarized light, and the rotation of the vibrational line of plane 
polarized light, are in every detail the same as we found in 211, 
212 for a medium wholly isotropic. But we shall see presently 
( 226 below) that two or three degrees of deviation from the axis 
produces a great change from the phenomena of a chiral isotropic 
medium ; and that for rays inclined 30 or more up to 90, the 
chirality is almost wholly swamped by the seolotropy, even when 
the seolotropy is as small as it is for quartz ( 223 below). As 
remarked in 220. there is no direction of light in a crystal, 
having three unequal values for , b, c, in which the chiro-optic 
effect is not masked by the seolotropy ; and therefore it is not 
so interesting to work out for a biaxal crystal the realised 
details of the solution (159), (162), (163). But it is exceedingly 
interesting to work them out for a uniaxal crystal, because of the 
great exaltation of the chiral phenomena when the ray is nearly in 
the direction of the axis, and because of the beautiful phenomena of 
Airy's spirals due to this exaltation ; and because of the admirable 
experimental investigation of the wave-surface in quartz crystal 
by McConnel referred to in 228 below. 

222. For waves transmitted in any direction through a 
uniaxal crystal; choose OZ as the axis, and therefore let p x = p y , 
and %* = %,,, in (158) (163) of 206. This reduces (163) to 
E = and therefore makes the waves in all directions real : that is 
to say transmissible with no change of motional configuration. 
Corresponding to (182) we may now put 



Py Pz 



OPPIC CHIRALITY IN QUARTZ CRYSTAL. 453 

Without loss of generality, we may simplify (158) (162) by Molar, 
taking 

^-'=0; X 2 =sin 8 0; i/ = cos 8 (190); 

with these, (163') of 206 gives 

(w 8 - a 1 ) [w 8 - a 8 + (a 3 - c 8 ) sin 8 0] = g [g - (g - h) sin 8 ff] u 8 . . .(191). 

This is a quadratic equation for the determination of the 
squares of the velocities of the two plane waves whose wave- 
normals are inclined at the same angle to OZ. For 6 = 90 we 
fall back on the case of 217, 218, 219, but with c instead of 6; 
and for = we fall back directly on 221. The exceedingly 
interesting transition from the subject of 221 to our present 
subject is fully represented by the solution of the quadratic 
equation for u s :-and is best explained by tables of values of the 
two roots from 9 = to 6 = 90 for some particular case or cases ; 
or graphic representation by curves as in figs. 17, 18, 19. 
I have chosen the case of quartz crystal traversed by sodium- 
light ; for which observation shows the rotation of the vibrational 
line to be 217 per centimetre of space travelled along the axis. 
This makes g= 4'605 . 10~ 5 . a; as we find by putting in the first 
member of (180), r = 217/360 : and in the last member u^ u^ = g; 
and a = '647593, the reciprocal of 1-54418, the smallest refractive 
index of quartz at 18, according to Rudberg; and r = '58932, the 
period in decimal of a michron, of the mean of sodium-lights 
D 1 D t ] the michron being the unit of time which makes the 
velocity of light unity when the unit of space is the michron (or 
millionth of a metre). (See footnote on p. 150 above.) 

As two sub-cases I have chosen h = and h = g ; because 
for all that our theory tells us, ( 223), hfg might be negative ; or 
might be zero or might have any positive value less than, or equal 
to, or greater than, unity. McConnel's experimental investigation 
seems ( 228 below) to make it certain that hjg for quartz is less 
than unity and probable that it is negative, and as small as 1, 
or perhaps smaller. As for g* t (189) shows that its value is 
inversely proportional to the square of the period of the light, 
if xx is the same for light of all periods ; g is positive or negative 
according as the crystal is optically right-handed or left-handed. 

223. c/a is the ratio of the smallest to the greatest refrac- 
tive index of quartz; that is 1'54418/1'55328, according to the 



454 LECTURE XX. 

Molar, figures used by McConnel ; being (as I see in Landolt and 
Bernstein's Tables) Rudberg's results for temperature 18 and 
sodium-light. From this we have (c/a) 2 = '98832 and (191) 
becomes 



.(192), 



or - (1 

a 4 *' i 

where 1 - w 2 = '01 1 68 sin 2 = ' 01 * 68 (1 - cos 20) . . . ( 1 93), 

and ^ = a ~ z g\.g (g h) sin 2 0] (194). 

If u^, u 2 2 denote the greater and less roots of (191), we have 

a~ 2 ( Ul * + ufi = I + w 2 + q (195); 

or 2 - w 2 2 ) = V[(l - w 2 ) + 2 (1 + w 2 ) q + g 2 ] . . . (196). 

Remark that when = 90, 1 - w 2 = '001168, q = a~ 2 gh. Hence 
a~ 2 gh might be positive, and as great as . 10~ 6 (1'168) 2 , or a 
little greater, without making the radical imaginary. 

224. In fig. 17, Curves 1 and 2 represent, from = to 
= 30, i [a~ 2 (^ 2 -uf)-(l- w 2 )] for the sub-cases h=g, and 
h = g ; and Curve 3 represents (1 w 2 ) on the same scale from 
= to = 5 0> 3. In fig. 18, Curves 1 and 2 represent, from 
6 = 30 to = 90, [or 2 (wj 2 w 2 2 ) (1 w-)] for the sub-cases 
h = g and h = g, on a scale of ordinates ten times, and abscissas 
half, that of fig. 17. 

Throughout the whole range of figs. 17 and 18, Curves 1 and 
2 represent chiral differences from the squares of the aeolotropic 
wave-velocities calculated according to the non-chiral constituent 
of the seolotropy of quartz crystal; Curve 3 of fig. 17, and 
equations (193) (196), show that through the range from = 
to 0= 5'3, the values of a" 1 ^, a~ l u z , and q differ from unity by 
less than 51 . 10~ 5 . Hence, through this range, we have, very 
approximately, 

i [a~ 2 (u? - M 2 2 ) - (1 - w 2 )] = Ul ~ u * _ (l - w ) ... (197). 



TWO TRIAL ASSUMPTIONS FOR QUARTZ COMPARED. 455 



456 



LECTURE XX. 



i O 
O 



MODIFICATION OF DOUBLE REFRACTION BYCHIRALITYIN QUARTZ. 457 

225. Fig. 19 illustrates the critical features of the crystal- Molar, 
line influence, and of the combined crystalline and chiral influence, 
of quartz-crystal on light traversing it with wave-normal inclined 
to the axis at any angle up to 14. The crystalline influence 
alone is represented in Curve 3 by downward ordinates equal to 
the excess of the velocity of the ordinary ray above the velocity 




of the extraordinary ray, divided by the former ; in an ideal crystal, 
corresponding to the mean of a right-handed and a left-handed 
quartz-crystal. Curve 1 represents the excess of the greater of 
the two wave-velocities in a real quartz-crystal above the velocity 
of the ordinary ray in the ideal mean crystal, divided by the 
latter. Curve 2 represents the excess (negative) of the less of 
the two wave-velocities in a real quartz-crystal above the velocity 
of the extraordinary ray in the ideal mean crystal, divided by the 
latter. According to our notation of 223, the ordinates of Curves 
3, 1, 2 are equal respectively to 

- 1 ; TT- 1 ; ?-* (!9 8 )- 



458 LECTURE XX. 

Molar. 226. From (198) we see that Curve 3 continued through the 
whole range from 6 = to 6 = 90, represents the distance between 
a tangent plane on a prolate ellipsoid of revolution of unit axial 
semi-diameter, and the parallel tangent plane on the circumscribed 
spherical surface of unit radius (the ellipsoid and the sphere 
constituting the wave-surface for the ideal uniaxal crystal cor- 
responding to the mean of right-handed and left-handed quartz). 
Curves 1 and 2 represent similarly the projections, outward from 
the sphere and inwards from the ellipsoid, of the two sheets of 
the wave-surface in a real quartz-crystal whether right-handed or 
left-handed. In each case 6 is the inclination to the equatorial 
plane, of the two tangent planes whose distance is represented by 
ordinates in the diagram. It is interesting to see and judge by 
Curve 1 how closely at 9 = 14, and thence up to 6 90, one of 
the sheets of the wave-surface in quartz agrees with the spherical 
surface: and to see by Curves 2 and 3 how nearly, at and above 14, 
. the other sheet agrees with the inscribed ellipsoid. On the other 
hand, it is interesting to see by the three curves how prepon- 
derating is chirality over aeolotropy, from 9 = to = 2 ; and 
to see how the preponderance gradually changes from chirality 
to seolotropy when 6 increases from 2 to 14. 

227. The characters of the two plane waves, whose wave- 
normals are inclined at angle to the axis of a quartz crystal, are 
to be discovered by commencing as in 222 ; and, for the case 
there defined, working out a realised solution from (159) of 206. 
We thus find that, for 6 = each wave is circularly polarized ; 
and, for all values of 6 between and 90, each wave is 
elliptically polarized ; the axes of the elliptic orbit being, one of 
them perpendicular to, and the other in, the plane through the 
wave-normal and the axis of the crystal. The former is the 
greater of the two axes for the wave which has the greater 
velocity (u^ ; the latter is the greater for the wave having the 
less velocity (u 2 ). For all values of greater than six or seven 
degrees the less axis of the elliptic orbit is very small in com- 
parison with the greater : that is to say each wave consists of very 
nearly rectilinear vibrations, or is very nearly " plane polarized " ; 
and one of the two waves approximates closely to the "ordinary ray," 
the other to the "extraordinary ray" in the ideal non-chiral crystal 
corresponding to the mean of right-handed and left-handed quartz. 



MATHEMATICAL THEORIES TESTED EXPERIMENTALLY. 459 

228. Looking at the two sub-cases represented by Curve 1 Molar, 
and Curve 2 of figures 17 and 18, we see that the difference 
between them is very small, probably quite imperceptible to the 
most delicate observation practicable, when 6 < 6. At 6 = 10 
the difference of the ordinates for the two curves is about 1/17 
of the ordinates for either ; and might be perceptible to observa- 
tion, though it represents an exceedingly small proportion, about 
1/500, of the whole difference of velocities between the two rays. 
This difference, as shown in figure 19, is itself very small, being 
only about 18.10~ s of a, the mean velocity of the right-handed and 
left-handed axial rays. How exceedingly searching McConneFs 
experimental investigation was may be judged by the fact shown 
in his figure 3, page 321*. that from = 14 to B = 30 he found 
definite systematic differences between " MacCullagh's Theory " 
and " Sarrau's Theory." The results of MacCullagh's Theory are 
expressed by our Curve 1 ; and our Curve 2 shows results differing 
from MacCullagh's in the same direction as Sarrau's but some- 
what more. Thus McConnel's investigation was more than amply 
sensitive to distinguish between our two sub-cases represented by 
Curves 1 and 2 of figures 17 and 18. Looking at McConnel's 
figure 3, and remarking that '003794 is the value of a b (our 
o c), which he took as correct according to his statement of 
Rudberg's results, we see that what he takes as Sarrau's Theory 
under-corrects the error of MacCullagh's : and our sub-case 2, 
which I chose on this account, must make the correction very 
nearly perfect. But I see on looking to Sarrau's paperf that 
his theory was not, as supposed by McConnel, confined to rela- 
tively small deviations from q = ^ cos 4 B (McConnel's notation of 

his page 314 translated into ours of 226); and that proper 
values of his constants (without the restriction of /, and g l 
relatively small, stated by McConnel), may be found to give a 
perfect agreement with McConnel's observations. The same may 
be said of Voigt's theory as pointed out by McConnel (p. 314). 
Thus we may take it that Sarrau's and Voigt's theories lead 
to results perfectly consistent with McConnel's observations. 
Theories of MacCullagh, Clebsch, Lang, and Boussinesq, are re- 
ferred to by McConnel as giving a constant for what we have 

* Phil. Trans. Roy. Soc., Part I., 1886. 

t Sarrau, Liouville, s4r. 2, tome xni. (1868), p. 101. 



460 



LECTURE XX. 



Molar, denoted by q, and as giving a fairly good approximation to the 
results of his observations ; but decidedly less good than Sarrau 
and Voigt, who allowed q to vary with 6. All these theories agree 
in taking the mutual forces between different parts of the vibrating 
medium as the origin of the chiral property, and differ essentially 
from my theory which ( 162166, 200205) finds it in virtual 
inertia of the ether as disturbed by chiral groups of atoms. 

229. The following Table shows, in degrees per centimetre, 
the rotation of the vibrational line of polarized light travelling 
through various substances; crystalline solid, and liquid; taken 
from Landolt and Bernstein's Tables, Edition 1894. 



Substance 


Quality 
of Light 


Direction 
of 
Rotation* 


Rotation of 
vibrational 
line in 
degrees per 
centimetre 


Observer 


Solid crystals : 
Cinnabar 


Red 




2700 to 3000 


Descloizeaux 


Quartz 


\ D (mean) 




217-27 ) 
217-05 


Sorettfe Sarasin 








216-84 J 




Lead Hyposulphate + 4 aq 


D 




55-31 


Pape 


Potash Hyposulphate ... 


D 




83-85 


Pape 


Potassium Sulphate 1 
Lithium Chromate 


D 




19-3 


H. Traube 


Sodium Bromate 


D 




21-7 


H. Traube 


Sodium Chlorate 


D 
D 




31-04 
31-6 


Guye 
Sohncke 


Sodium Periodate + 3 aq 


D 




233-0 


Groth 


Liquids : 










Turpentine C 10 H 16 
Nicotine C 10 H 14 N 2 


D 
D 


R. 
L. 


1-4147 xdt 
16-155 x df 


Landolt 
Landolt 













* Clockwise as seen by the observer is called right-handed (R.), anti-clockwise, 
left-handed (L.). 

t Where d denotes the specific gravity of the fluid. See p. 450 of Landolt and 
Bornstein. I do not see any good reason for the necessity of introducing 'd in the 
manner indicated in my table in the text, and rendered necessary by the notation 
adopted in the tables of Landolt and Bornstein. Some other embarrassing peculiari- 
ties in their notation have prevented me from venturing to quote any one of the 
numerous examples of the rotatory effect of " active " substances dissolved in 
" non-active" liquids given in pp. 450458 of these tables. 



FARADAY'S MAGNETO-OPTIC ROTATION. 461 

230. Important and interesting information regarding chiro- Molar, 
optic properties of liquids and solids is to be found in Mascart's 
Traite d'Optique, Volume n., Edition 1891, pages 247 to 369 ; and 
regarding Faraday's magneto-optic rotation of the vibrational line 
in pages 370 to 392. The first section of Appendix I in the 
present volume contains an important statement, given to me by 
the late Sir George Stokes, regarding chirality in crystals, and 
in crystalline molecules. Appendix H (Molecular Tactics of a 
Crystal, 22 footnote, 4752) contains statements of funda- 
mental principles in the pure geometry of chirality. Appendix G 
contains a complete mathematical theory of the quasi-inertia of 
a solid of any shape moving through a perfect liquid, with special 
remarks on chirality of this quasi-inertia. 

231. Lecture XX. as originally given, and fully reported in 
the papyrograph edition, and an appendix to it entitled "Improved 
Gyrostatic Molecule," contained unsatisfactory dynamical efforts to 
illustrate or explain Faraday's magneto-optic rotation. These are 
not reproduced in the present volume : but instead, an old paper 
of date 1856 entitled " Dynamical Illustration of the Magnetic 
and the Helic,oidal Rotatory Effects of Transparent Bodies on 
Polarized Light " is reproduced as Appendix F. This paper con- 
tains a statement of dynamical principles concerned in the two 
kinds of rotation ,of the vibrational line of plane polarized light 
travelling through transparent solids or fluids, which I believe 
may even now be accepted as fundamentally correct. When we 
have a true physical theory of the disturbance produced by a 
magnet in pure ether, and in ether in the space occupied by 
ponderable matter, fluid or solid, there will probably be no 
difficulty in giving as thoroughly satisfactory explanation of the 
magneto-optic rotation as we now have of the chiro-optic. 

232. In conclusion, let us consider what modification of the 
original Maxwell-Sellmeier dynamics of ordinary and anomalous 
dispersion must be made when we adopt the atomic hypothesis of 
Appendices A and E and of Lee. XIX., 162168. 

In App. A it is temporarily assumed, for the sake of a definite 
illustration, that the enormous variation of the etherial density 
within an atom is due to a purely Boscovichian force acting on 
the ether, in lines through the centre of the atom and varying 



462 LECTURE XX. 

Molecular, as a function of the distance. This makes no provision for 
vibrator or vibrators within an atom ; and, for the explanation of 
molecular vibrators, it only grants such molecular groups of atoms, 
as we have had for fifty years in the kinetic theory of gases, 
according to Clansius' impregnable doctrine of specific heats with 
regard to the partition of energy between translational and other 
than translational movements of the molecules. Now, in App. E, 
and in applications of it suggested in 162 168 of Lee. XIX., 
we have foundation for something towards a complete electro- 
etherial theory, of the Stokes- Kirchhoff vibrators* in the dynamics 
of spectrum-analysis, and of the Maxwell-Sellmeier explanation of 
dispersion. 

233. In our new theory, every single electrion within a 
mono-electrionic atom, and every group of two, three, or more, 
electrions, within a poly-electrionic atom, is a vibrator which, in 
a source of light, takes energy from its collision with other atoms, 
and radiates out energy in waves travelling through the sur- 
rounding ether. But at present we are not concerned with the 
source ; and in bringing this last of our twenty lectures to an 
end, I must limit myself to finding the effect of the presence of 
electrionic vibrators in ether, on the velocity of light traversing it. 

234. The " fundamental modes" of which, in Lee. X., p. 120, 
we have denoted the periods by K, K,, K,,, ... are now modes of 
vibration of the electrions within a fixed atom, when the ether 
around it and within it has no other motion than what is produced 
by vibrations of the electrions. It is to be remarked however that 
a steady motion of the atom through space occupied by the ether, 
will not affect the vibrations of the electrions within it, relatively 
to the atom. 

235. To illustrate, consider first the simple case of a mono- 
electrionic atom having a single electrion within it. There is just 
one mode of vibration, and its period is 



(199), 



where a denotes the radius of the atom, e the quantity of resinous 
electricity in an electrion, and m its virtual mass; and c denotes 

* See Lee. IX. pp. 101, 102, 103. 



ELECTRO-ETHERIAL EXPLANATION OF UNIFORM RIGIDITY. 463 

e*or*. This we see because the atom, being mono-electrionic, has Molecular, 
the same quantity of vitreous electricity as an electrion has of 
resinous ; and therefore (App. E, 4) the force towards the centre, 
experienced by an electrion held at a distance x from the centre, 
is e*cr a x ; which is denoted in 240 by car. 

236. Consider next a group of t electrions in equilibrium, 
or disturbed from equilibrium, within an t-electrionic atom. The 
force exerted by the atom on any one of the electrions is ie?Qr*D, 
towards the centre, if D is its distance from the centre. Let now 
the group be held in equilibrium with its constituents displaced 
through equal parallel distances, x, from their positions of equi- 
librium. Parallel forces each equal to ie 3 or 3 x, applied to the 
electrions, will hold them in equilibrium*; and if let go, they will 
vibrate to and fro in parallel lines, all in the same period 

\/^.. ...(200). 

V* e 

This therefore is one of the fundamental modes of vibration of 
the group ; and it is clearly the mode of longest period. Thus 
we see that the periods of the gravest vibrational modes of different 
electrionic vibrators are directly as the square roots of the cubes 
of the radii of the atoms and inversely as the square roots of the 
numbers of the electrions ; provided that in each case the atom 
is electrically neutralised by an integral number of electrions. 
Compare App. E, 6. 

237. I now propose an assumption which, while greatly 
simplifying the theory of the quasi inertia-loading of ether when 
it moves through space occupied by ponderable matter as set forth 
in App. A, perfectly explains the practical equality of the rigidity 
of ether through all space, whether occupied also by, or void of, 
ponderable matter. My proposal is that the radius of an electrion 
is so extremely small that the quantity of ether ivithin its sphere of 
condensation (Lee. XIX., 166) is exceedingly small in comparison 
with the quantity of 'undisturbed ether in a volume equal to the 
volume of the smallest atom. t 

This assumption, in connection with 164, 166 of Lee. XIX., 
makes the density of the ether exceedingly nearly constant through 

* Compare App. E, 23. 



464 LECTURE XX. 

Molecular, all space outside the spheres of condensation of electrions. This 
is true of space whether void of atoms, or occupied by closely 
packed, or even overlapping, atoms ; and the spheres of con- 
densation occupy but a very small proportion of the whole space 
even where most densely crowded with poly-electrionic atoms. 
The highly condensed ether within the sphere of condensation 
close around each electrion might have either greater or less 
rigidity than ether of normal density, without perceptibly marring 
the agreement between the normal rigidity of undisturbed ether, 
and the working rigidity of the ether within the atom. This 
seems to me in all probability the true explanation of what 
everyone must have felt to be one of the greatest difficulties in 
the dynamical theory of light ; the equality of the rigidity of 
ether inside and outside a transparent body. 

238. The smallness of the rarefaction of the ether within an 
atom and outside the sphere or spheres of condensation around its 
electrions, implies exceedingly small contribution to virtual inertia 
of vibrating ether, by that rarefaction ; so small that I propose to 
neglect it altogether. Thus if an atom is temporarily deprived of 
its electrion or electrions (rendering it vitreously electrified to the 
highest degree possible), ether vibrating to and fro through it will 
experience the same inertial resistance as if undisturbed by the 
atom. Its presence will not be felt in any way by the ether 
existing in the same place. Thus the actual inertia-loading of 
ether to which the refraction of light is due, is produced prac- 
tically by the electrions, and but little if at all perceptibly by 
the atoms, of the transparent body. 

239. For the present I assume an electrion to be massless, 
that is to say devoid of intrinsic inertia, and to possess virtual 
inertia only on account of the kinetic energy which accompanies 
its steady motion through still ether. This is in reality an energy 
of relative motion ; and does not exist when electrion and ether 
are moving at the same speed. See App. A passim, and equation 
(202) 240 below. 

240. Come now to the wave-velocity problem and begin 
with the simplest possible case, only one electrion in each atom. 
Consider waves of ^-vibration travelling y-wards according to the 
formula (203) below. Take a sample atom in the wave-plane at 
distance y from XOZ. The atom is unmoved by the ether- waves ; 



ELECTRO-ETHERIAL THEORY OF REFRACTION. 465 

while the electrion is set vibrating to and fro through its Molecular, 
centre. 

At time t, let x be the displacement of the electrion, from the 
centre of the atom (or its absolute displacement because at present 
we assume the atom to be absolutely fixed) : 

f the displacement of the ether around the atom : 

p the mean density of the ether within and around the atom, 
being, according to our assumptions, exactly the same as the 
normal density of undisturbed ether : 

n the rigidity of the ether within and around the atoms, being, 
according to our assumptions, very approximately the same at 
every point as the rigidity of undisturbed ether : 

N the number of atoms per unit of volume : 

ex the electric attraction towards the centre of its atom, 
experienced by the electrion in virtue of its displacement, x : 

ra the virtual mass of an electrion : 

E a cube of ether equal to l/N of the unit of volume, having 
the centre of one, and only one, atom within it. 

The equation of motion of E, multiplied by N, is 



and the equation of motion of the electrion within it, is 

' ** (202). 

241. The solution of these two equations for the regular 
regime of wave-motion is of the form 

in o> (t - 9} ; x = C' sin o>(t- &) (203), 



where a> is given. Our present object is to find the two un- 
knowns C/C' (or /#), and v. By (203) we see that 

* P; *, 2? (204). 

rft 3 ' dy 1 v 1 

This reduces (201) and (202) to 

-^*-tf(*-f) (205), 

T. L. 30 



466 LECTURE XX. 

Molecular, from which we find 

x mtf = -*_ 

2 2 2 



and = r + M =^+ - (207). 

fl 2 w n \ / n n r 2 -K 2 

The last member, is introduced with the notation 

T =!T ; *= 2 V? (208); 

where r denotes the period of the waves, and K the period of an 
electrion displaced from the centre of its atom, and left vibrating 
inside, while the surrounding ether is all at rest except for the 
outward travelling waves, by which its energy is carried away at 
some very small proportionate .rate per period ; perhaps not more 
than 10" 6 . It is clear that the greater the wave-length of the 
outgoing waves, in comparison with the radius of the sphere of 
condensation of the vibrating electrion, the smaller is the pro- 
portionate loss of energy per period. (Compare with the more 
complex problem, in which there are outgoing waves of two 
different velocities, worked out in the Addition to Lee. XIV., 
pp. 190 219. See particularly the examples in pp. 217, 218, 
219.) 

242. Look back now to the diagram of Lee. XII., p. 145, 
representing our complex molecular vibrator of Lee. L, pp. 12, 13, 
reduced to a single free mass, m ; connected by springs with the 
rigid sheath, the lining of an ideal spherical cavity in ether. In 
respect to that old diagram, let x now denote what was denoted 
on p. 145 by x ; that is to say the displacement of the ether, 
relatively to m. Thus in the old illustrative ideal mechanism, 
ex denotes a resultant force of springs acting on m : in the new 
suggestion of an electro-etherial reality ex denotes simply the 
electric attraction of the atom on its electrion w, when displaced 
to a distance x from its centre. In the old mechanism it is the 
pulls on ether by the springs, equal and opposite to their forces 
on m, by which ra acts on the ether (always admittedly an unreal 
kind of agency, invoked only by way of dynamical illustration). 
In the new electric design, in acts directly on the ether, in simple 
proportion to acceleration of relative motion. It does so because, 
in virtue of the ether's inertia when m is being relatively accelerated 



END OF LECTURE XX. 467 

the ether is less dense before than behind m, and therefore the Molecular, 
resultant of ra's attraction on it is backwards. 

It is interesting to see that every one of the formulas of 
240, 241 (with the new notation of x, in the old dynamical 
problem), are applicable to both the old and the new subjects: 
and to know that the -solution of the problem in terms of periods 
is the same in the two cases, notwithstanding the vast difference 
between the artificial and unreal details of the mechanism thought 
of and illustrated by models in 1884, and the probably real details 
of ether, electricity and ponderable matter, suggested in 1900 
1903. 

243. The interesting question of energy referred to in 
Lee. X., 11. 18 21 of p. Ill becomes more and more interesting 
now when we seem to understand its real quadruple character in 

(I) kinetic energy of pure ether, 

(II) potential energy of elasticity of ether, 

(III) electric potential energy of mutual repulsions of elec- 

trions and of attractions between electrions and atoms, 

(IV) potential energy of attraction of electrions on ether. 

It is slightly and imperfectly treated in App. C. It must, when 
fully worked out, include a dynamical theory of phosphorescence. 
For the present I must leave it with much regret, to allow this 
Volume to be prepared for publication. 



30-2 



APPENDIX A. 

ON THE MOTION PRODUCED IN AN INFINITE ELASTIC SOLID 
EY THE MOTION THROUGH THE SPACE OCCUPIED BY IT 
OF A BODY ACTING ON IT ONLY BY ATTRACTION OR 
REPULSION*. 

1. THE title of the present communication describes a 
pure problem of abstract mathematical dynamics, without in- 
dication of any idea of a physical application. For a merely 
mathematical journal it might be suitable, because the dynamical 
subject is certainly interesting both in itself and in its relation 
to waves and vibrations. My reason for occupying myself with 
it, and for offering it to the Royal Society of Edinburgh, is that 
it suggests a conceivable explanation of the greatest difficulty 
hitherto presented by the undulatory theory of light ; the 
motion of ponderable bodies through infinite space occupied by 
an elastic solid -f. 

2. In consideration of the confessed object, and for brevity, 
I shall use the word atom to denote an ideal substance occupying 
a given portion of solid space, and acting on the ether within it 
and around it, according to the old-fashioned eighteenth century 
idea of attraction and repulsion. That is to say, every infinitesimal 
volume A of the atom acts on every infinitesimal volume B of the 
ether with a force in the line PQ joining the centres of these two 
volumes, equal to 

Af(P,PQ) P B (1), 

* Communicated to the Phil. Mag. by the author, having heen read before the 
Eoyal Society of Edinburgh, July 16th, 1900; and before the "Congres" of the 
Paris Exhibition in August 1900. 

f The so-called "electro-magnetic theory of light" does not cut away this 
foundation from the old undulatory theory of light. It adds to that primary theory 
an enormous province of transcendent interest and importance ; it demands of us 
not merely an explanation of all the phenomena of light and radiant heat by 
transverse vibrations of an elastic solid called ether, but also the inclusion of 
electric currents, of the permanent magnetism of steel and lodestone, of magnetic 
force, and of electrostatic force, in a comprehensive etherial dynamics. 



FORCE BETWEEN ETHER AND ATOM. 469 

where p denotes the density of the ether at Q, and /(P, PQ) 
denotes a quantity depending on the position of P and on the 
distance PQ. The whole force exerted by the atom on the 
portion pB of the ether at Q, is the resultant of all the forces 
calculated according to (1), for all the infinitesimal portions A 
into which we imagine the whole volume of the atom to be 
divided. 

3. According to the doctrine of the potential in the well- 
known mathematical theory of attraction, we find rectangular 
components of this resultant as follows: 



...(2), 



where x, y, z denote coordinates of Q referred to lines fixed with 
reference to the atom, and < denotes a function (which we call 
the potential at Q due to the atom) found by summation as 
follows : 



(3), 



where fffA denotes integration throughout the volume of the 
atom. 

4. The notation of (1) has been introduced to signify that 
no limitation as to admissible law of force is essential ; but no 
generality, that seems to me at present practically desirable, is 
lost if we assume, henceforth, that it is the Newtonian law of the 
inverse square of the distance. This makes 



(4), 



and therefore drf(P,r) = -^^ (5), 

J PQ " V 

where a is a coefficient specifying for the point, P, of the atom, 
the intensity of its attractive quality for ether. Using (5) in (3) 
we find 

TQ <6>. 

and the components of the resultant force are still expressed 
by (2). We may suppose a to be either positive or negative 



470 APPENDIX A. 

(positive for attraction and negative for repulsion) ; and in fact 
in our first and simplest illustration of the problem we suppose 
it' to be positive in some parts and negative in other parts of the 
atom, in such quantities as to fulfil the condition 

SSSA* = ........................... (7). 

5. As a first and very simple illustration, suppose the atom 
to be spherical, of radius unity, with concentric interior spherical 
surfaces of equal density. This gives, for the direction of the 
resultant force on any particle of the ether, whether inside or 
outside the spherical boundary of the atom, a line through the 
centre of the atom. We may now take A = 4?rr 2 cr. The further 
assumption of (7) may thus be expressed by 



Q . ...(8); 

o 

and this, as we are now supposing the forces between every 
particle of the atom and every particle of the ether to be subject 
to the Newtonian law, implies, that the resultant of its attractions 
and repulsions is zero for every particle of ether outside the 
boundary of the atom. To simplify the case to the utmost, we 
shall further suppose the distribution of positive and negative 
density of the atom, and the law of compressibility of the ether, 
to be such, that the average density of the ether within the atom 
is equal to the undisturbed density of the ether outside. Thus 
the attractions and repulsions of the atom in lines through its 
centre produce, at different distances from its centre, condensa- 
tions and rarefactions of the ether, with no change of the total 
quantity of it within the boundary of the atom; and therefore 
produce no disturbance of the ether outside. To fix the ideas, 
and to illustrate the application of the suggested hypothesis to 
explain the refractivity of ordinary isotropic transparent bodies 
such as water or glass, I have chosen a definite particular case in 
which the distribution of the ether when at rest within the atom 
is expressed by the following formula, and partially shown in 
the accompanying diagram (fig. 1), and tables of calculated 
numbers : 

r 's 



Here, r' denotes the undisturbed distance from the centre of the 
atom, of a particle of the ether which -is at distance r when at 



ARBITRARY EXAMPLE OF DENSITY-DISTURBANCE. 471 

rest under the influence of the attractive and repulsive forces. 

According to this notation B (r 3 ) is the disturbed volume of a 
8 

spherical shell of ether whose undisturbed radius is r' and thick- 

A-. 

ness Br and volume -.- B (r' 9 ). Hence, if we denote the disturbed 

8 

and undisturbed densities of the ether by p and unity respectively, 
we have 

p8(r s ) = 8(r' 3 ) (10); 

whence, by (9), 



3 [l 



.(11). 



This gives 1 + K for the density of the ether at the centre 
of the atom. In order that the disturbance may suffice for 
refractivities such as those of air, or other gases, or water, or 
glass, or other transparent liquids or isotropic solids, according 
to the dynamical theory explained in 16 below, I find that K 
may for some cases be about equal to 100, and for others must 
be considerably greater. I have therefore taken K = 100, and 
calculated and drawn the accompanying tables and diagram 
accordingly. 

TABLE I. 



Col. 1 


Col. 2 


Col. 3 


Col. 3' 


Col. 4 


Col. 5 


r 1 


=i+*(i-rr 


r 


/-r 


P 


(P-V 


0-00 


101-0 


o-ooo 


o-ooo 


101-0 


o-ooo 


05 


91-25 


Oil 


039 


88-1 


on 


10 


82-0 


023 


077 


75-3 


039 


20 


65-0 


049 


151 


55-8 


132 


30 


50-0 


082 


218 


39-1 


256 


40 


37-0 


120 


280 


25-8 


357 


50 


26-0 


169 


331 


15-8 


423 


60 


17-0 


233 


367 


8-76 


423 


70 


10-0 


325 


375 


4-17 


338 


80 


5-0 


468 


332 


1-60 


131 


85 


3-25 


578 


272 


0-90 


-0-033 


90 


2-00 


715 


185 


0-50 


- -256 


95 


1-25 


865 


085 


35 


- '486 


96 


1-16 


897 


063 


36 


- -515 


97 


1-09 


928 


042 


39 


- -525 


-98 


1-04 


957 


023 


46 


- '496 


99 


1-01 


982 


008 


61 


- -376 


1-00 


1-00 


1-000 


000 


1-00 


- -ooo 



472 



APPENDIX A. 



TABLE II. 



Col. 1 


Col. 2 


Col. 3 


Col. 4 


Col. 5 


r 





r-r' 


p 


(/3-l)r 2 


o-oo 


o-ooo 


0-000 


101-00 


o-ooo 


02 


091 


071 


78-5 


030 


04 


169 


129 


64-4 


191 


06 


235 


175 


49-6 


175 


08 


297 


217 


39-5 


246 


10 


351 


251 


31-8 


308 


20 


551 


351 


11-8 


432 


30 


677 


377 


5-00 


360 


40 


758 


358 


2-46 


234 


50 


816 


316 


1-34 


085 


60 


858 


258 


0-82 


- 0-065 


70 


895 


195 


0-53 


- -231 


80 


929 


129 


0-38 


- -397 


90 


961 


061 


0-36 


- -518 


1-00 


1-000 


000 


1-00 


000 



6. The diagram (fig. 1) helps us to understand the dis- 
placement of ether and the resulting distribution of density, 
within the atom. The circular arc marked TOO indicates a 
spherical portion of the boundary of the atom ; the shorter of 

the circular arcs marked '95, '90, '20, '10 indicate spherical 

surfaces of undisturbed ether of radii equal to these numbers. 
The positions of the spherical surfaces of the same portions of 
ether under the influence of the atom, are indicated by the arc 
marked TOO, and the longer of the arcs marked '95, '90, ... '50, 
and the complete circles marked '40, '30, '20, '10. It may be 
remarked that the average density of the ether within any one 
of the disturbed spherical surfaces, is equal to the cube of the 
ratio of the undisturbed radius to the disturbed radius, arid is 
shown numerically in column 2 of Table I. Thus, for example, 
looking at the table and diagram, we see that the cube of the 
radius of the short arc marked '50 is 26 times the cube of the 
radius of the long arc marked '50, and therefore the average 
density of the ether within the spherical surface corresponding to 
the latter is 26 times the density (unity) of the undisturbed ether 
within the spherical surface corresponding to the former. The 
densities shown in column 4 of each table are the densities of the 
ether at (not the average density of the ether within) the con- 
centric spherical surfaces of radius r in the atom. Column 5 in 



DIAGRAM OF DENSITY-DISTURBANCE. 



473 



each table shows l/4rre of the excess (positive or negative) of the 
quantity of ether in a shell of radius r and infinitely small 
thickness e as disturbed by the atom above the quantity in a 



1-00 



95 



90 



80 



70 



60 



-50 



1-00 



95 



90 



50 



60 



50 



40 



30 




1-06 



95 



so 



60 



50 



Fig. 1. 



474 



APPENDIX A. 



shell of the same dimensions of undisturbed ether. The formula 
of col. 2 makes r = 1 when r' = 1, that is to say the total quantity 
of the disturbed ether within the radius of the atom is the same 
as that of undisturbed ether in a sphere of the same radius. 
Hence the sum of the quantities of ether calculated from col. 5 
for consecutive values of r, with infinitely small differences from 
r = to r=l, must be zero. Without calculating for smaller 
differences of r than those shown in either of the tables, we find 
a close verification of this result by drawing, as in fig. 2, a curve 
to represent (p l)r 2 through the points for which its value is 
given in one or other of the tables, and measuring the areas on 
the positive and negative sides of the line of abscissas. By 
drawing on paper (four times the scale of the annexed diagram), 
showing engraved squares of '5 inch and "1 inch, and counting 
the smallest squares and parts of squares in the two areas, I have 
verified that they are equal within less than 1 per cent, of either 
sum, which is as closq as can be expected from the numerical 
approximations shown in the tables, and from the accuracy 
attained in the drawing. 



Fig. 2. 

. 7. In Table I. (argument /) all the quantities are shown 
for chosen values of r', and in Table II. for chosen values of r. 



ATOM MOVING THROUGH SPACE OCCUPIED BY ETHER. 475 

The calculations for Table I. are purely algebraic, involving 
merely cube roots beyond elementary arithmetic. To calculate 
in terms of given values of r the results shown in Table II. 
involves the solution of a cubic equation. They have been 
actually found by aid of a curve drawn from the numbers of 
col. 3 Table I., showing r in terms of r'. The numbers in col. 2 
of Table II. showing, for chosen values of r, the corresponding 
values of r, have been taken from the curve ; and we may verify 
that they are approximately equal to the roots of the equation 
shown at the head of col. 2 of Table I., regarded as a cubic for r' 
with any given values of r and K. 

Thus, for example, taking r = '929 we calculate r = '811, 
,, / = -816 r = '498, 

r' = -677 r = -301, 

/ = -091 r=-0208, 

where we should have r = '8, '5, "3, and O2 respectively. These 
approximations are good enough for our present purpose. 

8. The diagram of fig. 2 is interesting, as showing how, 
with densities of ether varying through the wide range of from 
'35 to 101, the whole mass within the atom is distributed among 
the concentric spherical surfaces of equal density. We see by it, 
interpreted in conjunction with col. 4 of the tables, that from the 
centre to '56 of the radius the density falls from 101 to 1. For 
radii from '56 to 1, the values of (p I)^ decrease to a negative 
minimum of '525 at r = "93, and rise to zero at r = 1. The place 
of minimum density is of course inside the radius at which 
(p 1) r* is a minimum ; by cols. 4 and 3 of Table I., and cols. 4 
and 1 of Table II., we see that the minimum density is about '35, 
and at distance approximately '87 from the centre. 

9. Let us suppose now our atom to be set in motion through 
space occupied by ether, and kept in motion with a uniform 
velocity v. which we shall first suppose to be infinitely small in 
comparison with the propagational velocity of equivoluminal* 
waves through pure ether undisturbed by any other substance 
than that of the atom. The velocity of the earth in its orbit 

* That is to say, waves of transverse vibration, being the only kind of wave in 
an isotropic solid in which every part of the solid keeps its volume unchanged 
during the motion. See Phil. Mag., May, August, and October, 1899. 



476 APPENDIX A. 

round the sun being about 1/10,000 of the velocity of light, is 
small enough to give results, kinematic and dynamic, in respect 
to the relative motion of ether and the atoms constituting the 
earth closely in agreement with this supposition. According to 
it, the position of every particle of the ether at any instant is the 
same as if the atom were at rest ; and to find the motion produced 
in the ether by the motion of the atom, we have a purely 
kinematic problem of which an easy graphic solution is found 
by marking on a diagram the successive positions thus determined 
for any particle of the ether, according to the positions of the 
atom at successive times with short enough intervals between 
them, to show clearly the path and the varying velocity of the 
particle. 

10. Look, for example, at fig. 3, in which a semi-circum- 
ference of the atom at the middle instant of the time we are 
going to consider, is indicated by a semicircle C^AC , with 
diameter C C W equal to two units of length. Suppose the centre 
of the atom to move from right to left in the straight line C C m 
with velocity '1, taking for unit of time the time of travelling 1/10 
of the radius. Thus, reckoning from the time when the centre 
is at C , the times when it is at C 2 , C 5 , C lo , C 1S , C^ are 2, 5, 10, 
18, 20. Let Q' be the undisturbed position of a particle of ether 
before time 2 when the atom reaches it, and after time 18 when 
the atom leaves it. This implies that Q'C 2 = Q'C K = 1, and 
(7 2 G' 10 = C 10 C 18 = '8, and therefore C W Q' = '6. The position of the 
particle of ether, which when undisturbed is at Q', is found for 
any instant t of the disturbance as follows : 

Take C C = t/10; draw Q'C, and calling this r' find r' - r by 
formula (9), or Table I. or II. : in Q'C take Q'Q = r'-r. Q is 
the position at time t of the particle whose undisturbed position 
is Q'. The drawing shows the construction for t = 2, and t = 5, 
and t = l8. The positions at times 2, 3, 4, 5, ... 15, 16, 17, 18 
are indicated by the dots marked 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 
5, 6, 7, 8 on the closed curve with a corner at Q', which has been 
found by tracing a smooth curve through them. This curve, 
which, for brevity, we shall call the orbit of the particle, is 
clearly tangential to the lines Q'C 2 and QfC a . By looking to the 
formula (9), we see that the velocity of the particle is zero at 
the instants of leaving Q[ and returning to it. Fig. 4 shows the 



AN ABSOLUTE ORBIT AND ITS RELATIVE STREAM-LINE. 477 




478 APPENDIX A. 

particular orbit of fig. 3, and nine others drawn by the same 
method ; in all ten orbits of ten particles whose undisturbed 
positions are in one line at right angles to the line of motion of 
the centre of the atom, and at distances 0, '1, '2, ... '9 from it. 
All these particles are again in one straight line at time 10, being 
what we may call the time of mid-orbit of each particle. The 
numbers marked on the right-hand halves of the orbits are times 
from the zero of our reckoning; the numbers 1, 2, 3 ... etc. on 
the left correspond to times 11, 12, 13 ... of our reckoning as 
hitherto, or to times 1, 2, 3 ... after mid-orbit passages. Lines 
drawn across the orbits through 1, 2, 3 ... on the left, show 
simultaneous positions of the ten particles at times 1, 2, 3 after 
mid- orbit. The line drawn from 4 across seven of the curved 
orbits, shows for time 4 after mid-orbit, simultaneous positions 
of eight particles, whose undisturbed distances are 0, '1, ... '7. 
Remark that the 'orbit for the first of these ten particles is a 
straight line. 

11. We have thus in 10 solved one of the two chief 
kinematic questions presented by our problem: to find the 
orbit of a particle of ether as disturbed by the moving atom, 
relatively to the surrounding ether supposed fixed. The other 
question, to find the path traced through the atom supposed fixed 
while, through all space outside the atom, the ether is supposed 
to move uniformly in parallel lines, is easily solved, as follows : 
Going back to fig. 3, suppose now that instead of, as in 10, the 
atom moving from right to left with velocity '1 and the ether 
outside it at rest, the atom is at rest and the ether outside it is 
moving from left to right with velocity '1. Let '2, '3, '4, '5, '6, '7, 
'8, '9, 0, 1, '2, '3, '4, '5, '6, '7, '8 be the path of a particle of ether 
through the atom marked by seventeen points corresponding to 
the same numbers unaccented showing the orbit of the same 
particle of ether on the former supposition. On both suppositions, 
the position of the particle of ether at time 10 from our original 
era, ( 10), is marked 0. For times 11, 12, 13, etc., the positions 
of the particle on the former supposition are marked 1, 2, 3, 4, 5, 
6, 7, 8 on the left half of the orbit. The positions of the same 
particle on the present supposition are found by drawing from the 
points 1, 2, 3, ... 7, 8 parallel lines to the right, 1 '1, 2 '2, 3 '3, ... 
7 '7, 8 '8, equal respectively to 1, '2, -3, ... '7, '8 of the radius of 



ABSOLUTE ORBITS OF TEN PARTICLES OF ETHER. 



479 



the atom, being our unit of length. Thus we have the latter half 
of the passage of the particle through the atom ; the first half is 
equal and similar on the left-hand side of the atom. Applying 



76 5 




Fig. 4. 



5 43 



the same process to every one of the ten orbits shown in 
fig. 4, and to the nine orbits of particles whose undisturbed 
distances from the central line on the other side are '1, '2, ... '9, 
we find the set of stream-lines shown in fig. 5 (p. 480). The 
dots on these lines show the positions of the particles at times 



480 



APPENDIX A. 



0, 1, 2, ... 19, 20 of our original reckoning ( 10). The numbers 
on the stream-line of the particle whose undisturbed distance 
from the central line is '6 are marked for comparison with fig. 3. 




Fig. 5. 

The lines drawn across the stream-lines on the left-hand side of 
fig. 5 show simultaneous positions of rows of particles of ether 
which, when undisturbed, are in straight lines perpendicular to 



KINETIC ENERGY OF ETHER WITHIN A MOVING ATOM. 481 

the direction of motion. The quadrilaterals thus formed within 
the left-hand semicircle show the figures to which the squares 
of ether, seen entering from the left-hand end of the diagram, 
become altered in passing through the atom. Thus we have 
completed the solution of our second chief kinematic question. 

12. The first dynamic question that occurs to us, returning 
to the supposition of moving atom and of ether outside it at rest, 
is: What is the total kinetic energy (*) of the portion of the 
ether which at any instant is within the atom ? To answer it, 
think of an infinite circular cylinder of the ether in the space 
traversed by the atom. The time-integral from any era t = of 
the total kinetic energy of the ether in this cylinder is tic\ because 
the ether outside the cylinder is undisturbed by the motion of the 
atom according to our present assumptions. Consider any circular 
disk of this cylinder of infinitely small thickness e. After the 
atom has passed it, it has contributed to tic, an amount equal to 
the time-integral of the kinetic energies of all the orbits of small 
parts into which we may suppose it divided, and it contributes 
no more in subsequent time. Imagine the disk divided into 
concentric rings of rectangular cross-section edr'. The mass of 
one of these rings is Zirrdr'e because its density is unity; and 
all its parts move in equal and similar orbits. Thus we find that 
the total contribution of the disk amounts to 



(12), 



where fdsPfdt denotes integration over one-half the orbit of a 
particle of ether whose undisturbed distance from the central 
line is r' ; (because $ds*/dP is the kinetic energy of an ideal 
particle of unit mass moving in the orbit considered). Now the 
time-integral id is wholly made up by contributions of successive 
disks of the cylinder. Hence (12) shows the contribution per 
time e/q, q being the velocity of the atom ; and (tc being the 
contribution per unit of time) we therefore have 



(13). 



13. The double integral shown in (13) has been evaluated 
with amply sufficient accuracy for our present purpose by seem- 
ingly rough summations; firstly, the summations fdsf/dt for the 

T. L. 31 



482 



APPENDIX A. 



ten orbits shown in fig. 4, and secondly, summation of these sums 
each multiplied by dr'r. In the summations for each half-orbit, 
ds has been taken as the lengths of the curve between the con- 
secutive points from which the curve has been traced. This 
implies taking dt = l throughout the three orbits corresponding 
to undisturbed distances from the central line equal respectively 
to 0, '6, '8 ; and throughout the other semi-orbits, except for the 
portions next the corner, which correspond essentially to intervals 
each < 1. The plan followed is sufficiently illustrated by the 
accompanying Table III., which shows the whole process of 
calculating and summing the parts for the orbit corresponding 
to undisturbed distance '7. 

Table IV. shows the sums for the ten orbits and the products 
of each sum multiplied by the proper value of r', to prepare for 
the final .integration, which has been performed by finding the 
area of a representative curve drawn on conveniently squared 
paper as described in 6 above. The result thus found is '02115. 
It is very satisfactory to see that, within '1 per cent., this agrees 
with the simple sum of the widely different numbers shown in 
col. 3 of Table IV. 



TABLE III. 

Orbit r = ?. 



TABLE IV. 



ds 


ds 2 


dt 


ds*/dt 


006 


000036 


0-14 


000257 


137 


018769 


1-00 


018769 


112 


012544 


1-00 


012544 


077 


005929 


1-00 


005929 


050 


002500 


1-00 


002500 


048 


002304 


1-00 


002304 


050 


002500 


1-00 


002500 


052 


002704 


1-00 


002704 


Sum 


047507 



/ 


Ids^dt 


l.r'.Jd^/dt 





0818 


00000 


1 


0804 


00080 


2 


0781 


00156 


3 


0769 


00231 


4 


0722 


00289 


5 


0670 


00335 


6 


0567 


00310 


7 


0475 


00332 


8 


0310 


00248 


9 


0114 


00102 


Sum 


02113 



14. Using in (13) the conclusion of 13, and taking q = 1, 
we find 

K = 27r.-002115 (14). 

A convenient way of explaining this result is to remark that it is 



EXTRA INERTIA OF ETHER MOVING THROUGH A FIXED ATOM. 483 

634 of the kinetic energy 1- - (-I) 2 ) of an ideal globe of rigid 



matter of the same bulk as our atom, moving with the same 
velocity. Looking now at the definition of K in the beginning 
of 12, we may put our conclusion in words, thus: The dis- 
tribution of etherial density within our ideal spherical atom 
represented by (11) with K = 100, gives rise to kinetic energy 
of the ether within it at any instant, when the atom is moving 
slowly through space filled with ether, equal to '634 of the kinetic 
energy of motion with the same velocity through ideal void space, 
of an ideal rigid globe of the same bulk as the atom, and the 
same density as the undisturbed density of the ether. Thus if 
the atom, which we are supposing to be a constituent of real 
ponderable matter, has an inertia of its own equal to 7 per unit 
of its volume, the effective inertia of its motion through space 

occupied by ether will be ^*(7 + -634); the diameter of the 

atom being now denoted by s (instead of 2 as hitherto), and the 
inertia of unit bulk of the ether being still (as hitherto) taken as 
unit of inertia. In all that follows we shall suppose 7 to be very 
great, much greater than 10* ; perhaps greater than 10 13 . 

15. Consider now, as in 11 above, our atom at rest; and 
the ether moving uniformly in the space around the atom, and 
through the space occupied by the atom, according to the curved 
stream-lines and the varying velocities shown in fig. 5. The 
effective inertia of any portion of the ether containing the atom 
will be greater than the simple inertia of an equal volume of the 

ether by the amount ^ s* x '634. This follows from the well-known 

dynamical theorem that the total kinetic energy of any moving 
body or system of bodies is equal to the kinetic energy due to the 
motion of its centre of inertia, plus the sum of the kinetic energies 
of the motions of all its parts relative to the centre of inertia. 

16. Suppose now a transparent body solid, liquid, or 
gaseous to consist of an assemblage of atoms all of the same 
magnitude and quality as our ideal atom defined in 2, and 
with 7 enormously great as described in 14. The atoms may 
be all motionless as in an absolutely cold solid, or they may have 
the thermal motions of the molecules of a solid, liquid, or gas at 

312 



484 APPENDIX A. 

any temperature not so high but that the thermal velocities are 
everywhere small in comparison with the velocity of light. The 
effective inertia of the ether per unit volume of the assemblage 
will be exceedingly nearly the same as if the atoms were all 
absolutely fixed, and will therefore, by 15, be equal to 



(15), 



where N denotes the number of atoms per cubic centimetre of the 
assemblage, one centimetre being now our unit of length. Hence, 
if we denote by V the velocity of light in undisturbed ether, its 
velocity through the space occupied by the supposed assemblage 
of atoms will be 



(16). 



17. For example, let us take 7V= 4 x 10 20 * ; and, as I find 
suits the cases of oxygen and argon, s = 1'42 x 10~ 8 , which gives 

N ^ s? = '60 x 10~ 3 . The assemblage thus defined would, if con- 
densed one-thousand-fold, have '6 of its whole volume occupied by 
the atoms and '4 by undisturbed ether ; which is somewhat denser 
than the cubic arrangement of globes 

f space unoccupied = 1 = '4764 

and less dense than the densest possible arrangement 

f IT > 

space unoccupied = 1 ^ ^ = -2595 



Taking now JV^s 3 = "60 x 10~ 3 in (16), we find for the refractive 

index of our assemblage 1'00019, which is somewhat smaller than 
the refractive index of oxygen (T000273). By taking for K a larger 
value than 100 in (11), we could readily fit the formula to give, in 

* I am forced to take this very large number instead of Maxwell's 19 x 10 18 , 
as I have found it otherwise impossible to reconcile the known viscosities and the 
known condensations of hydrogen, oxygen, and nitrogen with Maxwell's theoretical 
formulas. [In 50 of Lect. XVII. of the present volume we saw that the smaller 
value 10 20 is admissible and probably may be not far from the truth.] It must be 
remembered that Avogadro's law makes N the same for all gases. 



DIFFICULTY RAISED BY MICHELSON's EXPERIMENT. 485 

an assemblage in which '6 x 10~* of the whole space is occupied 
by the atom, exactly the refractive index of oxygen, nitrogen, or 
argon, or any other gas. It is remarkable that according to the 
particular assumptions specified in 5, a density of ether in the 
centre of the atom considerably greater than 100 times the 
density of undisturbed ether is required to make the refractivity 
as great as that of oxygen. There is, however, no difficulty in 
admitting so great a condensation of ether by the atom, if we 
are to regard our present problem as the basis of a physical 
hypothesis worthy of consideration. 

18. There is, however, one serious, perhaps insuperable, 
difficulty to which I must refer in conclusion : the reconciliation 
of our hypothesis with the result that ether in the earth's 
atmosphere is motionless relatively to the earth, seemingly proved 
by an admirable experiment designed by Michelson, and carried 
out with most searching care to secure a trustworthy result, by 
himself and Morley*. I cannot see any flaw either in the idea 
or in the execution of this experiment. But a possibility of 
escaping from the conclusion which it seemed to prove may be 
found in a brilliant suggestion made independently by Fitzgerald f, 
and by LorentzJ of Leyden, to the effect that the motion of ether 
through matter may slightly alter its linear dimensions; according 
to which if the stone slab constituting the sole plate of Michelson 
and Morley 's apparatus has, in virtue of its motion through space 
occupied by ether, its lineal dimensions shortened one one-hundred- 
millionth in the direction of motion, the result of the experiment 
would not disprove the free motion of ether through space occupied 
by the earth. 

* Phil. Mag., December, 1887. 

t Public Lectures in Trinity College, Dublin. 

J Versuch einer Theorie der electrischen und optiachen Ertcheinungen in bewegten 
Korpern. Leiden, 1895. 

This being the square of the ratio of the earth's velocity round the sun 
(30 kilometres per sec.) to the velocity of light (300,000 kilometres per sec.). 



APPENDIX B. 

NINETEENTH CENTURY CLOUDS OVER THE DYNAMICAL 
THEORY OF HEAT AND LIGHT*. 

(Friday evening Lecture, Royal Institution, April 27, 1900.) 

[!N the present article the substance of the lecture is repro- 
duced with large additions, in which work commenced at the 
beginning of last year and continued after the lecture, during 
thirteen months up to the present time, is described with results 
confirming the conclusions and largely extending the illustrations 
which were given in the lecture. I desire to take this opportunity 
of expressing my obligations to Mr William Anderson, my secretary 
and assistant, for the mathematical tact and skill, the accuracy of 
geometrical drawing, and the unfailingly faithful perseverance in 
the long-continued and varied series of drawings and algebraic 
and arithmetical calculations, explained in the following pages. 
The whole of this work, involving the determination of results 
due to more than five thousand individual impacts, has been 
performed by Mr Anderson. K., Feb. 2, 1901.] 

1. The beauty and clearness of the dynamical theory, which 
asserts heat and light to be modes of motion, is at present ob- 
scured by two clouds. I. The first came into existence with the 
undulatory theory of light, and was dealt with by Fresnel and 
Dr Thomas Young; it involved the question, How could the earth 
move through an elastic solid, such as essentially is the lumini- 
ferous ether ? II. The second is the Maxwell-Boltzmann doctrine 
regarding the partition of energy. 

2. CLOUD I. RELATIVE MOTION OF ETHER AND PONDER- 
ABLE BODIES; such as movable bodies at the earth's surface, 
stones, metals, liquids, gases ; the atmosphere surrounding the 
earth; the earth itself as a whole; meteorites, the moon, the sun, 

* Journal of the Royal Institution. Also Phil. Mag. July, 1901. 



MOTION OF ETHER THROUGH PONDERABLE MATTER. 487 

and other celestial bodies. We might imagine the question satis- 
factorily answered, by supposing ether to have practically perfect 
elasticity for the exceedingly rapid vibrations, with exceedingly 
small extent of distortion, which constitute light; while it behaves 
almost like a fluid of very small viscosity, and yields with ex- 
ceedingly small resistance, practically no resistance, to bodies 
moving through it as slowly as even the most rapid of the 
heavenly bodies. There are, however, many very serious objections 
to this supposition; among them one which has been most noticed, 
though perhaps not really the most serious, that it seems in- 
compatible with the known phenomena of the aberration of light. 
Referring to it, Fresnel, in his celebrated letter* to Arago, wrote 
as follows: 

" Mais il parait impossible d'expliquer 1'aberration des 6toiles 
"dans cette hypothese; je n'ai pu jusqu'a present du moins con- 
"cevoir nettement ce phenomene qu'en supposant que I'e'ther 
" passe librement au travers du globe, et que la vitesse communi- 
" que"e a ce fluide subtil n'est qu'une petite partie de celle de la 
" terre; n'en excede pas le centieme, par exemple. 

"Quelque extraordinaire que paraisse cette hypothese au premier 
" abord, elle n'est point en contradiction, ce me semble, avec I'id6e 
" que les plus grands physiciens se sont faite de 1'extreme porosite" 
" des corps." 

The same hypothesis was given by Thomas Young, in his 
celebrated statement that ether passes through among the mole- 
cules or atoms of material bodies like wind blowing through a 
grove of trees. It is clear that neither Fresnel nor Young had 
the idea that the ether of their undulatory theory of light, with 
its transverse vibrations, is essentially an elastic solid, that is to 
say, matter which resists change of shape with permanent or 
sub-permanent force. If they had grasped this idea they must 
have noticed the enormous difficulty presented by the laceration 
which the ether must experience if it moves through pores or 
interstices among the atoms of matter. 

3. It has occurred to me that, without contravening any- 
thing we know from observation of nature, we may simply deny 
the scholastic axiom that two portions of matter cannot jointly 

* Annales de Chimie, 1818 ; quoted in full by Larnior in his recent book, Mtlier 
and Matter, pp. 320 322. 



488 APPENDIX B. 

occupy the same space, and may assert, as an admissible hypo- 
thesis, that ether does occupy the same space as ponderable 
matter, and that ether is not displaced by ponderable bodies 
moving through space occupied by ether. But how then could 
matter act on ether, and ether act on matter, to produce the 
known phenomena of light (or radiant heat), generated by the 
action of ponderable bodies on ether, and acting on ponderable 
bodies to produce its visual, chemical, phosphorescent, thermal, 
and photographic effects ? There is no difficulty in answering 
this question if, as it probably is, ether is a compressible and 
dilatable* solid. We have only to suppose that the atom exerts 
force on the ether, by which condensation or rarefaction is pro- 
duced within the space occupied by the atom. At present f I 
confine myself, for the sake of simplicity, to the suggestion of a 
spherical atom producing condensation and rarefaction, with con- 
centric spherical surfaces of equal density, but the same total 
quantity of ether within its boundary as the quantity in an 
equal volume of free undisturbed ether. 

4. Consider now such an atom given at rest anywhere in 
space occupied by ether. Let force be applied to it to cause it to 
move in any direction, first with gradually increasing speed, and 
after that with uniform speed. If this speed is anything less 
than the velocity of light, the force may be mathematically proved 
to become zero at some short time after the instant when the 
velocity of the atom becomes uniform, and to remain zero for 
ever thereafter. What takes place is this : 

5. During all the time in which the velocity of the atom is 
being augmented from zero, two sets of non-periodic waves, one 
of them equi-voluminal, the other irrotational (which is therefore 
condensational-rarefactional), are being sent out in all directions 
through the surrounding ether. The rears of the last of these 
waves leave the atom, at some time after its acceleration ceases. 
This time, if the motion of the ether outside the atom, close 

* To deny this property is to attribute to ether indefinitely great resistance 
against forces tending to condense it or to dilate it which seems, in truth, an 
infinitely difficult assumption. 

f Further developments of the suggested idea have been contributed to the 
Royal Society of Edinburgh, and to the Congres International de Physique held in 
Paris in August. (Proc. R.S.E. July 1900 ; Vol. of reports, in French, of the Cong. 
Inter,; and Phil. Mag., Aug., Sept., 1900.) 



UNIFORM MOTION OF ATOM RESISTED IF QUICKER THAN LIGHT. 489 

beside it, is infinitesimal, is equal to the time taken by the slower 
wave (which is the equi-voluminal) to travel the diameter of the 
atom, and is the short time referred to in 4. When the rears 
of both waves have got clear of the atom, the ether within it and 
in the space around it, left clear by both rears, has come to a 
steady state of motion relatively to the atom. This steady motion 
approximates more and more nearly to uniform motion in parallel 
lines, at greater and greater distances from the atom. At a 
distance of twenty diameters it differs exceedingly little from 
uniformity. 

6. But it is only when the velocity of the atom is very 
small in comparison with the velocity of light, that the dis- 
turbance of the ether in the space close round the atom is 
infinitesimal. The propositions asserted in 4 and the first sen- 
tence of 5 are true, however little the final velocity of the atom 
falls short of the velocity of light. If this uniform final velocity 
of the atom exceeds the velocity of light, by ever so little, a 
non-periodic conical wave of equi-voluminal motion is produced, 
according to the same principle as that illustrated for sound by 
Mach's beautiful photographs of illumination by electric spark, 
showing, by changed refractivity, the condensational-rarefactional 
disturbance produced in air by the motion through it of a rifle 
bullet. The semi-vertical angle of the cone, whether in air or 
ether, is equal to the angle whose sine is the ratio of the wave 
velocity to the velocity of the moving body*. 

* On the same principle we see that a body moving steadily (and, with little 
error, we may say also that a fish or water-fowl propelling itself by fins or web-feet) 
through calm water, either floating on the surface or wholly submerged at some 
moderate distance below the surface, produces no wave disturbance if its velocity is 
less than the minimum wave velocity due to gravity and surface tension (being about 
23 cms. per second, or -44 of a nautical mile per hour, whether for sea water or fresh 
water); and if its velocity exceeds the minimum wave velocity,. it produces a wave 
disturbance bounded by two lines inclined on each side of its wake at angles each 
equal to the angle whose sine is the minimum wave velocity divided by the velocity 
of the moving body. It is easy for anyone to observe this by dipping vertically a 
pencil or a walking-stick into still water in a pond (or even in a good-sized hand- 
basin), and moving it horizontally, first with exceeding small speed, and afterwards 
faster and faster. I first noticed it nineteen years ago, and described observations 
for an experimental determination of the minimum velocity of waves, in a letter to 
William Froude, published in Nature for Oct., in Phil. Mag. for Nov. 1871, and 
in App. O below, from which the following is extracted. "[Recently, in the 
" schooner yacht Lalla Rookh], being becalmed in the Sound of Mull, I had an 
" excellent opportunity, with the assistance of Professor Helmholtz, and my brother 



490 APPENDIX B. 

7. If, for a moment, we imagine the steady motion of the 
atom to be at a higher speed than the wave velocity of the 
condensational-rarefactional wave, two conical waves, of angles 
corresponding to the two wave velocities, will be steadily pro- 
duced ; but we need not occupy ourselves at present with this 
case because the velocity of the condensational-rarefactional wave 
in ether is, we are compelled to believe, enormously great in 
comparison with the velocity of light. 

8. Let now a periodic force be applied to the atom so as to 
cause it to move to and fro continually, with simple